Abstract
We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic problem which is well-posed in the Sobolev space \(H^r({\mathbb {S}})\) for each \(r\in (2,3)\). When neglecting surface tension effects, the Muskat problem is a fully nonlinear evolution equation and of parabolic type in the regime where the Rayleigh–Taylor condition is satisfied. We then establish the well-posedness of the Muskat problem in the open subset of \(H^2({\mathbb {S}})\) defined by the Rayleigh–Taylor condition. Besides, we identify all equilibrium solutions and study the stability properties of trivial and of small finger-shaped equilibria. Also other qualitative properties of solutions such as parabolic smoothing, blow-up behavior, and criteria for global existence are outlined.
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1 Introduction and the Main Results
In this paper we study the coupled system of equations
for \(t> 0 \)Footnote 1 and \( x\in {\mathbb {R}},\) which is supplemented by the initial condition
The evolution problem (1.1) describes the motion of the boundary \([y=f(t,x)+tV]\) separating two immiscible fluid layers with unbounded heights located in a homogeneous porous medium with permeability \(k\in (0,\infty )\) or in a vertical/horizontal Hele–Shaw cell. It is assumed that the fluid system moves with constant velocity (0, V), \(V\in {\mathbb {R}}\), that the motion is periodic with respect to the horizontal variable x (with period \(2\pi \)), and that the fluid velocities are asymptotically equal to (0, V) far away from the interface. The unknowns of the evolution problem (1.1) are the functions \((f,{\overline{\omega }})=(f,{\overline{\omega }})(t,x)\). We denote by \({\mathbb {S}}:={\mathbb {R}}/2\pi {\mathbb {Z}}\) the unit circle, functions that depend on \(x\in {\mathbb {S}}\) being \(2\pi \)-periodic with respect to the real variable x. To be concise, we have set
and \((\,\cdot \,)'\) denotes the spatial derivative \(\partial _x.\) We further denote by g the Earth’s gravity, \(\sigma \in [0,\infty )\) is the surface tension coefficient, \(\kappa (f(t))\) is the curvature of the free boundary \([y=f(t,x)+tV]\), while \(\mu _\pm \) and \(\rho _\pm \) are the viscosity and the density, respectively, of the fluid ± which occupies the unbounded periodic strip
Moreover, the real constant \(\Theta \) and the Atwood number \(a_\mu \) that appear in (1.1a)\(_2\) are defined by
The integrals in (1.1a) are singular at \(s=0\) and \(\mathrm{PV}\) denotes the Cauchy principle value. In this paper we consider a general setting where
The observation that \(|a_\mu |<1\) is crucial for our analysis. This property enables us to prove, for suitable f(t), that the Eq. (1.1a)\(_2\) has a unique solution \({\overline{\omega }}(t)\) (which depends in an intricate way on f(t), see Sects. 4 and 5). Therefore we shall only refer to f as being the solution to (1.1).
The Muskat problem, in the classical formulation (2.1), dates back to M. Muskat’s paper [52] from 1934. However, many of the mathematical studies on this topic are quite recent and they cover various physical scenarios and mathematical aspects related to the original model proposed in [52], cf. [6, 7, 9, 11,12,13,14,15,16,17,18, 21,22,23,24,25, 27, 30, 32, 36, 38,39,43, 48, 49, 49, 53, 54, 58, 60,61,62] (see also [55, 56] for some recent research on the compressible analogue of the Muskat problem, the so-called Verigin problem).
Below we discuss only the literature pertaining to (1.1) and its nonperiodic counterpart. In the presence of surface tension effects, that is for \(\sigma >0\), (1.1) has been studied previously only in [7] where the author proved well-posedness of the problem in \(H^r\) (with \(r\ge 6\)) in the more general setting of interfaces which are parameterized by curves, and the zero surface tension limit of the problem has been also considered there. The nonperiodic counterpart to (1.1) has been investigated in [48] where it was shown that the problem is well-posed in \(H^r({\mathbb {S}})\) for each \(r\in (2,3)\) by exploiting the fact that the problem is quasilinear parabolic together with the abstract theory outlined in [4, 5] for such problems. Additionally, it was shown in [48] that the problem exhibits the effect of parabolic smoothing and criteria for global existence of solutions were found. We showed herein that the results in the nonperiodic framework [48] hold also for (1.1). Besides, this paper provides the first full picture of the set of equilibrium solutions to (1.1)—which are described by either flat of finger-shaped interfaces—and the stability properties of the flat equilibria and of small finger-shaped equilibria are studied in the natural phase space \(H^r({\mathbb {S}})\). For the latter purpose we use a quasilinear principle of linearized stability established recently in [50].
The first main result of this paper is the following theorem establishing the well-posedness of the Muskat problem with surface tension in the setting of classical solutions and for general initial data together with other qualitative properties of the solutions.
Theorem 1.1
Let \(\sigma >0\) and \(r\in (2,3) \) be given. Then, the following hold:
-
(i)
(Well-posedness in \(H^r({\mathbb {S}})\)) The problem (1.1) possesses for each \(f_0\in H^r({\mathbb {S}})\) a unique maximal solution
$$\begin{aligned} f:=f(\cdot ; f_0)\in \mathrm{C}([0,T_+(f_0)),H^r({\mathbb {S}}))\cap \mathrm{C}((0,T_+(f_0)), H^3({\mathbb {S}}))\cap \mathrm{C}^1((0,T_+(f_0)), L_2({\mathbb {S}})), \end{aligned}$$with \(T_+(f_0)\in (0,\infty ]\), and \([(t,f_0)\mapsto f(t;f_0)]\) defines a semiflow on \(H^r({\mathbb {S}})\).
-
(ii)
(Global existence/blow-up criterion) If
$$\begin{aligned} \sup _{[0,T_+(f_0))\cap [0,T]}\Vert f(t;f_0)\Vert _{H^r}<\infty \qquad \text {for all }T>0, \end{aligned}$$then \(T_+(f_0)=\infty \).
-
(iii)
(Parabolic smoothing) The mapping \([(t,x)\mapsto f(t,x)]:(0,T_+(f_0))\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is real-analytic. In particular, f(t) is a real-analytic function for all \(t\in (0,T_+(f_0))\).
Remark 1.2
-
(i)
Despite that we deal with a third order problem in the setting of classical solutions, the curvature of the initial data in Theorem 1.1 may be unbounded and/or discontinuous. Moreover, it becomes instantaneously real-analytic under the flow.
-
(ii)
Solutions which are not global have, in view of Theorem 1.1, the property that
$$\begin{aligned} \sup _{[0,T_+(f_0)) }\Vert f(t)\Vert _{H^s}=\infty \qquad \text {for each } s\in (2,3). \end{aligned}$$
Concerning the stability of equilibria, we also have to differentiate between the cases \(\sigma =0\) and \(\sigma >0\). Before doing this we point out two features that are common for both cases. Firstly, the integral mean of the solutions to (1.1) (found in Theorems 1.1 or 1.5 below) is constant with respect to time, see Sect. 6. Secondly, (1.1) has the following invariance property: If f is a solution to (1.1), then the translation
is also a solution to (1.1). For these two reasons, we shall only address the stability issue for equilibria to (1.1) which have zero integral mean and under perturbations with zero integral mean. However, because of the invariance property (1.2), our stability results can be transferred also to other equilibria, see Remark 1.4.
To set the stage, let
In Theorem 1.3 below we describe the stability properties of some of the equilibria to (1.1) when \(\sigma >0\). In this case the equilibrium solutions to (1.1) are either constant functions or finger-shaped as in Fig. 1. The finger-shaped equilibria exist only in the regime where \(\Theta <0\), that is when either the fluid located below has a larger density or when the less viscous fluid advances into the region occupied by the other one with sufficiently high speed |V|. Furthermore, these equilibria form global bifurcation branches (see Sect. 6 for the complete picture of the set of equilibria).
Theorem 1.3
Let \(\sigma >0\) and \(r\in (2,3)\) be given. The following hold:
-
(i)
If \(\Theta +\sigma >0\), then \(f=0\) is exponentially stable. More precisely, given
$$\begin{aligned} \omega \in (0,k(\sigma +\Theta )/(\mu _-+\mu _+)), \end{aligned}$$there exist constants \(\delta >0\) and \(M>0\), with the property that if \(f_0\in \widehat{H}^r({\mathbb {S}})\) satisfies \(\Vert f_0\Vert _{H^{r}}\le \delta \), the solution to (1.1) exists globally and
$$\begin{aligned} \Vert f(t;f_0)\Vert _{H^r } \le Me^{-\omega t}\Vert f_0\Vert _{H^{r} }\qquad \text {for all }t\ge 0. \end{aligned}$$ -
(ii)
If \(\Theta +\sigma <0\), then \(f=0\) is unstable. More precisely, there exists \(R>0\) and a sequence \((f_{0,n})\subset \widehat{H}^{r}({\mathbb {S}})\) of initial data such that:
-
\(f_{0,n}\rightarrow 0 \) in \(\widehat{H}^{r}({\mathbb {S}});\)
-
There exists \(t_n\in (0,T_+(f_{0,n}))\) with \(\Vert f(t_n;f_{0,n})\Vert _{H^r}=R\).
-
-
(iii)
(Instability of small finger shaped equilibria) Given \(1\le \ell \in {\mathbb {N}}\), there exists a real-analytic bifurcation curve \((\lambda _\ell ,f_\ell ):(-\varepsilon _\ell ,\varepsilon _\ell )\rightarrow (0,\infty )\times \widehat{H}^3({\mathbb {S}})\), \(\varepsilon _\ell >0\), with
$$\begin{aligned} \left\{ \begin{array}{lll} \lambda _\ell (s)=\ell ^2-\frac{3\ell ^4}{8}s^2+O(s^4) \quad \text {in }{\mathbb {R}},\\ f_\ell (s)= s\cos (\ell x)+O(s^2) \quad \text {in } \widehat{H}^3({\mathbb {S}}) \end{array} \right. \qquad \text {for }s\rightarrow 0, \end{aligned}$$such that \(f_{\ell }(s)\) is an even equilibrium to (1.1) if \(\Theta =-\sigma \lambda _\ell (s)\). The finger-shaped equilibrium \(f_\ell (s)\), \(0<|s|<\varepsilon _\ell \), is unstable if \(\varepsilon _\ell \) is sufficiently small in the sense there exists \(R>0\) and a sequence \((f_{0,n})\subset \widehat{H}^{r}({\mathbb {S}})\) such that:
-
\(f_{0,n}\rightarrow f_\ell (s) \) in \(\widehat{H}^{r}({\mathbb {S}});\)
-
There exists \(t_n\in (0,T_+(f_{0,n}))\) with \(\Vert f(t_n;f_{0,n})-f_{\ell }(s)\Vert _{H^r}=R\).
-
With respect to Theorem 1.3 we add the following remarks (Remark 1.4 (i) remains valid for Theorem 1.6 below as well).
Remark 1.4
-
(i)
If f is an even equilibrium to (1.1), the translation \(f(\cdot -a)+c\), \(a,\, c\in {\mathbb {R}}\), is also an equilibrium solution. In fact, all equilibria can be obtained in this way (see Sect. 6). The invariance property (1.2) shows that f and \(f(\cdot -a)+c\) have the same stability properties.
-
(ii)
It is shown in Theorem 6.1 that the local curves \((\lambda _\ell ,f_\ell )\) can be continued to global bifurcation branches consisting entirely of equilibrium solutions to (1.1). The stability issue for the large finger-shaped equilibria remains an open problem.
When switching to the regime where \(\sigma =0\), many aspects in the analysis of the Muskat problem with surface tension have to be reconsidered. A first major difference to the case \(\sigma >0\) is due to the fact that the quasilinear character of the problem, which is mainly due to the curvature term, is lost (except for the very special case when \(\mu _-=\mu _+\), cf. [47]), and the problem (1.1) is now fully nonlinear. The second important difference, is that the problem is of parabolic type only when the Rayleigh–Taylor condition holds. The Rayleigh–Taylor condition originates from [59] and is expressed in terms of the pressures \(p_\pm \) associated of the fluid ± as follows
with \(\nu \) denoting the unit normal to the curve \([y=f_0(x)]\) pointing towards \(\Omega _+^V(0)\) . The first result in this setting is a local existence result in \(H^k({\mathbb {S}})\), with \(k\ge 3\), established in [21] in the more general setting of interfaces parametrized by periodic curves (for initial data such that the Rayleigh–Taylor conditions holds). The particular case of fluids with equal densities has been in fact investigated previously in [60] and the authors have shown the existence of global solutions for small data. The methods from [21] have been then generalized in [22] to the three-dimensional case, the analysis leading to a local existence result in \(H^k\) with \(k\ge 4\). More recently in [15] the authors have established global existence and uniqueness of solutions to (1.1) for small data in \(H^2({\mathbb {S}})\) together with some exponential decay estimates in \(H^r\)-norms with \(r\in [0,2)\). For the nonperiodic Muskat problem with \(\sigma =0\) it is moreover shown in [15] there exist unique local solutions for initial data in \(H^2({\mathbb {R}})\) which are small in the weaker \(H^{3/2+\varepsilon }\)-norm with \(\varepsilon \in (0,1)\) arbitrarily small. The latter smallness size condition on the data was dropped in [48] where it is shown that the nonperiodic Muskat possesses for initial data in \(H^2({\mathbb {R}})\) that satisfy the Rayleigh–Taylor condition a unique local solution and that the solution depends continuously on the data. Lastly, we mention the paper [37] where the existence and uniqueness of a weaker notion of solutions is established for the nonperiodic Muskat problem with initial data in critical spaces, together with some algebraic decay of the global solutions. In this paper we first generalize the methods from the nonperiodic setting [48] to prove the well-posedness of (1.1) for general initial data in \(H^2({\mathbb {S}})\) and instantaneous parabolic smoothing for solutions which satisfy an additional bound. Before presenting our result, we point out that if \(\Theta =0\), then (1.1) has only constant solutions for each \(f_0\in H^r({\mathbb {R}})\), with \(r>3/2\), as Theorem 3.3 shows that in this case \({\overline{\omega }}=0\) is the only solution to (1.1a)\(_2\) that lies in \(\widehat{L}_2({\mathbb {S}}).\) When \(\Theta \ne 0\), the situation is much more complex. Letting
denote the set of initial data in \(H^2({\mathbb {S}})\) for which the Rayleigh–Taylor condition holds, it is shown in Sect. 5 that \({\mathcal {O}}\) is nonempty precisely when \(\Theta >0\). This condition on the constants has been identified also in the nonperiodic case. In fact, we prove that if \(\Theta >0\), then \({\mathcal {O}}\) is an open subset of \(H^2({\mathbb {S}})\) which contains all constant functions. Using the abstract fully nonlinear parabolic theory established in [26, 46], we prove below that the Muskat problem without surface tension is well-posed in the set \({\mathcal {O}}\), cf. Theorem 1.5. Physically, in the particular situation when gravity is neglected \(\Theta >0\) is equivalent to the fact that the more viscous fluid enters the region occupied by less viscous one, while in the case \(V=0\) the condition \(\Theta >0\) means that the fluid located below has a larger density.
Theorem 1.5
Let \(\sigma =0\), \(\mu _-\ne \mu _+\),Footnote 2 and assume that \(\Theta >0\). Given \(f_0\in {\mathcal {O}}\), the problem (1.1) possesses a solution
for some \(T>0\) and an arbitrary \(\alpha \in (0,1)\). Additionally, the following statements are true:
-
(i)
f is the unique solution to (1.1) belonging to
$$\begin{aligned} \bigcup _{\beta \in (0,1)} C([0,T],{\mathcal {O}})\cap C^1([0,T], H^1({\mathbb {S}}))\cap C^{\beta }_{\beta }((0,T], H^2({\mathbb {S}})). \end{aligned}$$ -
(ii)
f may be extended to a maximally defined solution
$$\begin{aligned} f(\,\cdot \,; f_0)\in C([0,T_+(f_0)),{\mathcal {O}})\cap C^1([0,T_+(f_0)), H^1({\mathbb {S}}))\cap \bigcap _{\beta \in (0,1)}C^{\beta }_{\beta }((0,T], H^2({\mathbb {S}})) \end{aligned}$$for all \(T<T_+(f_0)\), where \(T_+(f_0)\in (0,\infty ].\)
-
(iii)
The solution map \([(t,f_0)\mapsto f(t;f_0)]\) defines a semiflow on \({\mathcal {O}}\) which is real-analytic in the open set \(\{(t,f_0)\,:\, f_0\in {\mathcal {O}},\, 0<t<T_+(f_0)\}\).
-
(iv)
If \(f(\,\cdot \,; f_0):[0,T_+(f_0))\cap [0,T]\rightarrow {\mathcal {O}}\) is uniformly continuous for all \(T>0\), then either \(T_+(f_0)=\infty \), or
$$\begin{aligned} T_+(f_0)<\infty \, \text { and }\, \mathrm{dist}(f(t;f_0),\partial {\mathcal {O}})\rightarrow 0\text { for }t\rightarrow T_+(f_0). \end{aligned}$$ -
(v)
If \(f(\,\cdot \,; f_0)\in B((0,T), H^{2+\varepsilon }({\mathbb {S}}))\) for some \(T\in (0,T_+(f_0))\) and \(\varepsilon \in (0,1)\) arbitrary small, then
$$\begin{aligned} f\in C^\omega ((0,T)\times {\mathbb {R}},{\mathbb {R}}). \end{aligned}$$
The assertions of Theorem 1.5 are weaker compared to that of Theorem 1.1. For example the uniqueness claim at (i) is established in the setting of strict solutions (in the sense of [46, Chapter 8]) which belong additionally to some singular Hölder space
with \(\beta \in (0,1).\) This drawback results from the fact that in the absence of surface tension effects we deal with a fully nonlinear (and nonlocal) problem. We also point out that the parabolic smoothing property established at (v) holds only for solutions \(f(\,\cdot \,; f_0)\in B((0,T), H^{2+\varepsilon }({\mathbb {S}}))\) for some \(\varepsilon >0\). This additional boundedness condition is needed because the space-time translation
does not define for \(a,\,b >0\) a bounded operator between these singular Hölder spaces. This property hiders us to use the parameter trick from the proof of Theorem 1.1 to establish parabolic smoothing for all solutions in Theorem 1.5. However, the boundedness hypothesis imposed at (v) is satisfied if \(f_0\in {\mathcal {O}}\cap H^3({\mathbb {S}})\) because the statements (i)–(iv) in Theorem 1.5 remain true when replacing \(H^k({\mathbb {S}})\) by \(H^{k+1}({\mathbb {S}}) \) for \(k\in \{1,2\}\) (possibly with a smaller maximal existence time).
Finally, we point out that in the case when \(\sigma =0\) the equilibrium solutions to (1.1) are the constant functions. Theorem 1.6 states that the zero solution to (1.1) (and therewith all other equilibria) is exponentially stable under perturbations with zero integral mean.
Theorem 1.6
(Exponential stability). Let \(\sigma =0\) and \(\Theta >0\). Then, given \(\omega \in (0,k\Theta /(\mu _-+\mu _+))\), there exist constants \(\delta >0\) and \(M>0\), with the property that if \(f_0\in \widehat{H}^2({\mathbb {S}})\) satisfies \(\Vert f_0\Vert _{H^{2}}\le \delta ,\) then \(T_+(f_0)=\infty \) andFootnote 3
Before proceeding with our analysis we emphasize that the periodic case considered herein is more involved that the “canonical” nonperiodic Muskat problem because abstract results from harmonic analysis, cf. [51, Theorem 1], which directly apply to the nonperiodic case (in order to establish useful mapping properties and commutator estimates) have no correspondence in the set of periodic functions. However, we derive in Appendix A, by using the results from the nonperiodic case [48, 49], the boundedness of certain multilinear singular integral operators which can be directly applied in the proofs. A further drawback of the Eq. (1.1a) is that some of the integral terms are of lower order and some of the arguments are therefore lengthy. Finally, we point out that the stability issue remains an open question for the nonperiodic counterpart of (1.1).
2 The Equations of Motion and the Equivalence of the Formulations
In this section we present the classical formulation of the Muskat problem (see (2.1) below) introduced in [52] and prove that this formulation is equivalent to the contour integral formulation (1.1) in a quite general setting, cf. Proposition 2.3.
We first introduce the equations of motion. In the fluid layers the dynamic is governed by the equations
where \(v_\pm (t):=(v_\pm ^1(t),v_\pm ^2(t))\) denotes the velocity field of the fluid \(\pm .\) While (2.1a)\(_1\) is the incompressibility condition, the Eq. (2.1a)\(_2\) is known as Darcy’s law. This linear relation is frequently used for flows which are laminar, cf. [10]. These equations are supplemented by the following boundary conditions at the free interface
where \(\nu (t) \) is the unit normal at \([y=f(t,x)+tV]\) pointing into \(\Omega _+^V(t)\) and \(\langle \, \cdot \,|\,\cdot \,\rangle \) the inner product in \({\mathbb {R}}^2\). Additionally, we impose the far-field boundary condition
The motion of the free interface is described by the kinematic boundary condition
and, since we consider \(2\pi \)-periodic flows, f(t), \(v_\pm (t)\), and \(p_\pm (t)\) are assumed to be \(2\pi \)-periodic with respect to x for all \(t\ge 0\). Finally, we supplement the system with the initial condition
It is convenient to rewrite the Eq. (2.1) in a reference frame that moves with the constant velocity (0, V). To this end we let
and
Direct computations show that (2.1) is equivalent to
In Proposition 2.3 we establish the equivalence of the two formulations (1.1) and (2.3). It is important to point out that the function \({\overline{\omega }}\) in (1.1a)\(_1\) is uniquely identified by f in the space \(\widehat{L}_2({\mathbb {S}})\) (this feature is established rigorously only later on in Theorem 3.3). This aspect is essential at several places in this paper, see Proposition 2.3 and the preparatory lemma below.
Lemma 2.1
Given \(f\in H^1({\mathbb {S}})\) and \( {\overline{\omega }} \in \widehat{L}_2({\mathbb {S}})\) let
for \((x,y)\in {\mathbb {R}}^2 {\setminus }[y=f(x)] \) and set \(V:=(V^1,V^2)\) and \(V_\pm :=V|_{\Omega _\pm },\) where
Then, there exists a constant \(C=C(\Vert f\Vert _\infty )>0\) such that
for all \((x,y)\in \Omega _\pm \) satisfying \(|y|\ge 1+2\Vert f\Vert _\infty \).
Proof
Let first \(f\ne 0\). Taking advantage of
for \(|y|\ge 2\Vert f\Vert _\infty ,\) it follows that
In order to estimate \(V^1_\pm \) we use the fact that \(\langle {\overline{\omega }}\rangle =0\) to derive, after performing some elementary estimates, that
for all \(|y|\ge 2\Vert f\Vert _\infty \). The claim for \(f=0\) follows in a similar way. \(\square \)
In Proposition 2.3 we show that, given a solution to (1.1), the velocity field in the classical formulation (2.1) at time t can be expressed in terms of \(f:=f(t)\) and \({\overline{\omega }}:={\overline{\omega }}(t)\) according to Lemma 2.1, provided that f and \({\overline{\omega }}\) have suitable regularity properties. We point out that a formal derivation of the formula (2.4) is provided, in a more general context, in [21, Section 2]. In Lemma 2.2 we establish further properties of the velocity field defined in Lemma 2.1.
Lemma 2.2
Let \(f\in H^2({\mathbb {S}})\) and \( {\overline{\omega }} \in \widehat{H}^1({\mathbb {S}})\). The vector field \(V_\pm \) introduced in Lemma 2.1 belongs to \(\mathrm{C}(\overline{\Omega }_\pm )\cap \mathrm{C}^1({\Omega _\pm })\), it is divergence free and irrotational, and
Letting further
for \( (x,y)\in \overline{\Omega }_\pm \), where \(c_\pm \in {\mathbb {R}}\) and \(d>\Vert f\Vert _\infty \), it holds that \( P_\pm \in \mathrm{C}^1(\overline{\Omega }_\pm )\cap \mathrm{C}^2({\Omega _\pm })\) and the relations (2.3)\(_1\)–(2.3)\(_3\), (2.3)\(_5\) are all satisfied.
Proof
The theorem on the differentiation of parameter integrals shows that \(V_\pm \) is continuously differentiable in \(\Omega _\pm ,\) divergence free, and irrotational. In order to show that \(V_\pm \in \mathrm{C}(\overline{\Omega }_\pm )\) it suffices to show that the one-sided limits when approaching a point \((x_0,f(x_0))\in [y=f(x)]\) from \(\Omega _-\) and \(\Omega _+\), respectively, exist. To this end we note that the complex conjugate of \((V^1_\pm ,V^2_\pm )\) satisfies
with \(\Gamma \) being a \(2\pi \)-period of the graph \([y=f(x)]\) and with \(g:\Gamma \rightarrow {\mathbb {C}}\) defined by
Given \(z=(x,y)\not \in [y=f(x)]\), it is convenient to write
because Lebesgue’s theorem now shows that if \(z_n\) approaches \(z_0=(x_0,f(x_0))\) from \(\Omega _+\) (or \(\Omega _-\)), then
Moreover, using Plemelj’s formula, cf. e.g. [45, Theorem 2.5.1], we find that
where the \(\mathrm{PV}\) is taken at \(\xi =z_0\), and we conclude that
The formula (2.5) and the property \(V_\pm \in \mathrm{C}(\overline{\Omega }_\pm )\) follow at once. The remaining claims are simple consequences of Lemma 2.1 and of the already established properties. \(\square \)
Using Lemmas 2.1 and 2.2, we conclude this section with the following equivalence result.
Proposition 2.3
(Equivalence of formulations). Let \(T\in (0,\infty ] \) be given.
-
(a)
Let \(\sigma =0\). The following are equivalent:
-
(i)
the problem (2.3) for \( f\in \mathrm{C}^1([0,T), L_2({\mathbb {S}}))\) and
$$\begin{aligned} \bullet&\, \, \, \, f(t)\in H^2({\mathbb {S}}),\, \, {\overline{\omega }}(t):=\big \langle (V_-(t)-V_+(t))|_{[y=f(t,x)]} \big |(1,f'(t))\big \rangle \in \widehat{H}^1({\mathbb {S}}),\\ \bullet&\, \, \, V_\pm (t)\in \mathrm{C}(\overline{\Omega }_\pm (t))\cap \mathrm{C}^1({\Omega _\pm (t)}), \, P_\pm (t)\in \mathrm{C}^1(\overline{\Omega }_\pm (t))\cap \mathrm{C}^2({\Omega _{\pm }(t)}) \end{aligned}$$for all \(t\in [0,T)\);
-
(ii)
the evolution problem (1.1) for \(f\in \mathrm{C}^1([0,T), L_2({\mathbb {S}}))\), \(f(t)\in H^2({\mathbb {S}})\), and \({\overline{\omega }}(t)\in \widehat{H}^1({\mathbb {S}})\) for all \(t\in [0,T)\).
-
(i)
-
(b)
Let \(\sigma >0\). The following are equivalent:
-
(i)
the problem (2.3) for \(f\in \mathrm{C}^1((0,T), L_2({\mathbb {S}}))\cap \mathrm{C}([0,T), L_2({\mathbb {S}}))\) and
$$\begin{aligned} \bullet&\, \, \, f(t)\in H^4({\mathbb {S}}), \,\, {\overline{\omega }}(t):=\big \langle (V_-(t)-V_+(t))|_{[y=f(t,x)]} \big |(1,f'(t))\big \rangle \in \widehat{H}^1({\mathbb {S}}), \\ \bullet&\, \, \, V_\pm (t)\in \mathrm{C}(\overline{\Omega }_\pm (t))\cap \mathrm{C}^1({\Omega _\pm (t)}), \, P_\pm (t)\in \mathrm{C}^1(\overline{\Omega }_\pm (t))\cap \mathrm{C}^2({\Omega _{\pm }(t)}) \end{aligned}$$for all \(t\in (0,T);\)
-
(ii)
the Muskat problem (1.1) for \(f\in \mathrm{C}^1((0,T), L_2({\mathbb {S}}))\cap \mathrm{C}([0,T), L_2({\mathbb {S}}))\), \( f(t)\in H^4({\mathbb {S}})\), and \( {\overline{\omega }}(t)\in \widehat{H}^1({\mathbb {S}}) \) for all \(t\in (0,T)\).
-
(i)
Proof
To prove the implication \((i)\Rightarrow (ii)\) of (a), let \((f,V_\pm ,P_\pm )\) be a solution to (2.3) on [0, T) and choose \(t\ge 0\) fixed but arbitrary (the time dependence is not written explicitly in this proof). Letting
denote the vorticity associated to the global velocity field
where \(\mathbf{1}_{\Omega _\pm }\) is the characteristic function of \(\Omega _\pm \), it follows from (2.3)\(_3\) and Stokes’ theorem that
where
Similarly as in the particular case \(\mu _-=\mu _+\), cf. [47, Proposition 2.2], we find that the global velocity field \((V^1,V^2)\) is given by (2.4). Lemma 2.2 now shows, together with the kinematic boundary condition, that f solves the Eq. (1.1a)\(_1.\) Besides, differentiating the Laplace–Young Eq. (2.3)\(_4\), the relations (2.3)\(_2\) and (2.5) finally lead us to (1.1a)\(_2,\) and the proof of this implication is complete.
For the reverse implication, we define \(V_\pm \) according to (2.4), and the pressures by (2.6). For suitable \(c_\pm \), it follows from (1.1a)\(_2\) and Lemmas 2.1, 2.2 that indeed \((f,V_\pm ,P_\pm )\) solves (2.3).
The equivalence stated at (b) follows in a similar way. \(\square \)
3 The Double Layer Potential and Its Adjoint
We point out that the Eq. (1.1a)\(_2\) is linear with respect to \({\overline{\omega }}(t)\). The main goal of this section is to address the solvability of this equation for \({\overline{\omega }}(t)\) in suitable function spaces, cf. Theorems 3.3 and 3.5. To this end we first associate to (1.1a) two singular operators and study their mapping properties (see Lemmas 3.1 and 3.2). Finally, in Theorem 3.6 and Lemma 3.7 we study the properties of the adjoints of these singular operators.
To begin, we write (1.1a)\(_2\) in the more compact form
where \({\mathbb {A}}(f)\) is the linear operator
Given \(f\in H^r({\mathbb {S}})\) with \(r>3/2\), we prove in Lemma 3.2 that \({\mathbb {A}}(f)\in {\mathcal {L}}(L_2({\mathbb {S}})).\) Then, it is a matter of direct computation to verify that \({\mathbb {A}}(f)\) is the \(L_2\)-adjoint of the double layer potential
A main part of the subsequent analysis is devoted to the study of the invertibility of the linear operator \(1+a_\mu {\mathbb {A}}(f) \) in the algebras \({\mathcal {L}}(\widehat{L}_2({\mathbb {S}}))\) and \({\mathcal {L}}(\widehat{H}^1({\mathbb {S}}))\). These invertibility properties enable us to solve (3.1) and to formulate (1.1) as an evolution equation for f only, that is
where we have associated to (1.1a)\(_1\) the operator \({\mathbb {B}}(f)\) defined by
As a first result we establish the following mapping properties.
Lemma 3.1
Given \(r>3/2\), it holds that
Proof
Let us first assume that
Given \(f,\, {\overline{\omega }}\in \mathrm{C}^\infty ({\mathbb {S}})\) with \(\langle {\overline{\omega }}\rangle =0\) let \(V_-\) be as defined in Lemma 2.1. Observing that
Stokes’ formula together with Lemmas 2.1, 2.2 yields
and therefore \({\mathbb {B}}(f)[{\overline{\omega }}]\in \widehat{L}_2({\mathbb {S}})\). This immediately implies (3.6).
Hence, we are left to establish (3.7). To this end it is convenient to write
where
Taking advantage of the relations
it is easy to see that \({\mathbb {B}}_i(f) \in {\mathcal {L}}(L_2({\mathbb {S}}), L_\infty ({\mathbb {S}}))\) for \(i\in \{1,2\}\) (and that \(\mathrm{PV}\) is not needed). In fact these mappings are real-analytic, that is
Furthermore, given \(\tau \in (1/2,1)\), classical (but lengthy) arguments (see [47, Lemmas 3.2-3.3] where similar integral operators are discussed) show that
and we are left to consider the operator \({\mathbb {B}}_3.\)
Recalling Lemma A.1, we see that
and Lemma A.1 (i) immediately yields \({\mathbb {B}}_3(f)\in {\mathcal {L}}(L_2({\mathbb {S}})).\) Moreover, arguing as in [49, Section 5], it follows that
In order to prove that \({\mathbb {B}}_3(f)\in {\mathcal {L}}(H^1({\mathbb {S}})),\) when additionally \(f\in H^2({\mathbb {S}})\), we let \(\{\tau _\varepsilon \}_{\varepsilon \in {\mathbb {R}}}\) denote the \(C_0\)-group of right translations, that is \(\tau _\varepsilon h(x)=h(x-\varepsilon )\) for \(x\in {\mathbb {R}}\) and \(h\in L_2({\mathbb {S}})\). Given \(\varepsilon >0\) and \({\overline{\omega }}\in H^1({\mathbb {S}})\), it holds that
Since
we may pass, in view of Lemma A.1 (i)–(iii), to the limit \(\varepsilon \rightarrow 0\) in the identity above to conclude that
in \(L_2({\mathbb {S}}).\) This proves that \({\mathbb {B}}_3(f)[{\overline{\omega }}]\in H^1({\mathbb {S}}),\) with
Lemma A.1 and the arguments in [49, Section 5] finally lead us to
and (3.7) follows now from (3.9), (3.11) and (3.13). This completes the proof. \(\square \)
We now study the mapping properties of the operator \({\mathbb {A}}\) introduced in (3.2).
Lemma 3.2
Let \(r>3/2\) be given. It then holds
Proof
Pick first \(f,\, {\overline{\omega }}\in \mathrm{C}^\infty ({\mathbb {S}})\) with \(\langle {\overline{\omega }}\rangle =0\) and let \(V_-\in \mathrm{C}(\overline{\Omega }_-)\cap \mathrm{C}^1(\Omega _-)\) be as defined in Lemma 2.1. It then holds
and therefore \({\mathbb {A}}(f)[{\overline{\omega }}]\in \widehat{L}_2({\mathbb {S}})\) if and only if \(\langle V_-|_{[y=f(x)]}|(1,f')\rangle \in \widehat{L}_2({\mathbb {S}})\). The latter property follows from the periodicity of f and \(P_-\), where \(P_-\in \mathrm{C}^1(\overline{\Omega }_-)\) is given in (2.6), with respect to x and the relation
We are thus left to prove that
We proceed as in the previous lemma and write
where, using the notation introduced in Lemma A.1, we have
Similarly as in Lemma 3.1, we get
with
The properties (3.9), (3.10), and (3.17) combined imply (3.15), and the proof is complete. \(\square \)
We now address the solvability of Eq. (3.1). To this end we first establish the invertibility of \(1+a_\mu {\mathbb {A}}(f) \) in \({\mathcal {L}}(\widehat{L}_2({\mathbb {S}}))\). A similar property has been established, under stronger assumptions on f which are not compatible with our approach, in [21].
Theorem 3.3
Let \(r>3/2\) and \(M>0\). Then, there exists a constant \(C=C(M)>0\) such that
for all \(\lambda \in {\mathbb {R}}\) with \(|\lambda |\ge 1,\)\({\overline{\omega }}\in \widehat{L}_2({\mathbb {S}}),\) and \(f\in H^r({\mathbb {S}})\) with \(\Vert f'\Vert _{\infty }\le M.\)
In particular, \(\{\lambda \in {\mathbb {R}}\,:\,|\lambda |\ge 1\}\) is contained in the resolvent set \({\mathbb {A}}(f)\in {\mathcal {L}}(\widehat{L}_2({\mathbb {S}}))\) for each \(f\in H^r({\mathbb {S}})\).
Proof
In view of Lemma 3.2, it suffices to establish the estimate (3.19) for \({\overline{\omega }},\, f\in \mathrm{C}^\infty ({\mathbb {S}})\) with \(\langle {\overline{\omega }}\rangle =0\) and \(\Vert f'\Vert _{\infty }\le M.\) Let \(V_\pm \in \mathrm{C}(\overline{\Omega }_\pm )\cap \mathrm{C}^1(\Omega _\pm )\) be as defined in Lemma 2.1 and set
We denote by \(\tau \) and \(\nu \) the tangent and the outward normal unit vectors at \(\partial \Omega _-\) and we decompose \(F_\pm \) in tangential and normal components \(F_\pm =F_\pm ^\tau +F_\pm ^\nu ,\) where
cf. (2.5). Recalling the Lemmas 3.1, 3.2, we may view \(F_\pm ^\tau \) and \(F^\nu _\pm \) as being elements of \( L_2({\mathbb {S}},{\mathbb {R}}^2).\)
We next introduce the bilinear form \({\mathcal {B}}:L_2({\mathbb {S}},{\mathbb {R}}^2)\times L_2({\mathbb {S}},{\mathbb {R}}^2)\rightarrow {\mathbb {R}}\) by the formula
for \( F=(F^1,F^2),\, G=(G^1,G^2)\in L_2({\mathbb {S}},{\mathbb {R}}^2).\) Inserting the vector fields \(F_\pm \) in (3.20), we find by using Lebesgue’s dominated convergence theorem, Stokes’ formula, and the Lemmas 2.1, 2.2 that
where \(\Gamma \) denotes again a period of the graph \([y=f(x)].\) Moreover, in virtue of (3.21), we may write (3.22) equivalently as
and, recalling that \(\Vert f'\Vert _{\infty }\le M\), we infer from (3.23) that
with a positive constant \(C=C(M)\). In particular we get
Given \(\lambda \in {\mathbb {R}}\) with \(|\lambda |\ge 1,\) it holds that
and eliminating the mixed term on the right hand side we obtain together with (3.23) that
from where we conclude that
with a constant \(C=C(M).\) The latter estimate and (3.24) yield (3.19). That \(\{\lambda \in {\mathbb {R}}\,:\,|\lambda |\ge 1\}\) belongs to the resolvent set of \({\mathbb {A}}(f)\in {\mathcal {L}}(\widehat{L}_2({\mathbb {S}}))\) for all \(f\in H^r({\mathbb {S}})\) is a straightforward consequence of (3.19), Lemma 3.2, and of the continuity method, cf. e.g. [5, Proposition I.1.1.1]. \(\square \)
The following remark is relevant in Sect. 6 in the stability analysis of the Muskat problem.
Remark 3.4
The estimate
derived in (3.24) enables us to identify the equilibrium solutions to the Muskat problem (1.1) (see (3.4)) as being the solutions to the capillarity equation
We now establish the invertibility of \(1+a_\mu {\mathbb {A}}(f) \) in the algebra \({\mathcal {L}}(\widehat{H}^1({\mathbb {S}}))\) under the assumption that \(f\in H^2({\mathbb {S}})\).
Theorem 3.5
Let \(M>0\). Then, there exists a constant \(C=C(M)>0\) such that
for all \(\lambda \in {\mathbb {R}}\) with \(|\lambda |\ge 1,\)\({\overline{\omega }}\in \widehat{H}^1({\mathbb {S}}),\) and \(f\in H^2({\mathbb {S}})\) with \(\Vert f\Vert _{H^2}\le M.\)
In particular, \(\{\lambda \in {\mathbb {R}}\,:\,|\lambda |\ge 1\}\) is contained in the resolvent set of \({\mathbb {A}}(f)\in {\mathcal {L}}(\widehat{H}^1({\mathbb {S}}))\) for each \(f\in H^2({\mathbb {S}})\).
Proof
Recalling (3.19), we are left to estimate the term \(\Vert ((\lambda -{\mathbb {A}}(f))[{\overline{\omega }}])'\Vert _{2} \) suitably. To this end, we infer from (3.16) and (3.18) that
where the operator \(T^A_{\mathrm{lot}}(f)\) defined by
encompasses all lower order terms of \(({\mathbb {A}}_3(f)[{\overline{\omega }}])'\) with respect to \({\overline{\omega }}\) as, for each \(\tau \in (1/2,1)\) fixed, it holds
with \(C=C(M)\). Indeed, letting \(r:=(9-2\tau )/4,\) it follows that \(r\in (3/2,2) \) and \(\tau \in (5/2-r,1)\), and the estimates (A.1) and (A.3) yield
Moreover, it follows from (3.9), (3.10) and the compactness of the embedding \(H^2({\mathbb {S}})\hookrightarrow H^r({\mathbb {S}})\) that also
Finally, using integration by parts in the formulas defining \({\mathbb {B}}_1(f) \) and \({\mathbb {B}}_2(f)\) we get
and (3.29) follows.
Invoking (3.19) and (3.29) we find a constant \(c=c(M)\in (0,1)\) with
and since by (3.30)Footnote 4 and Young’s inequality
for some \(C'=C'(M),\) it follows that
This estimate together with (3.19) leads us to (3.26) and the proof is complete. \(\square \)
We conclude this section by considering the adjoints of the operators defined in (3.2) and (3.5). Firstly we establish a similar estimate as in Theorem 3.5 for the operator \(P({\mathbb {A}}(f))^*,\) where \(({\mathbb {A}}(f))^*\) is the double layer potential, cf. (3.3), and where \(P:L_2({\mathbb {S}})\rightarrow \widehat{L}_2({\mathbb {S}})\), with \(Ph:=h-\langle h\rangle ,\) denotes the orthogonal projection on \(\widehat{L}_2({\mathbb {S}})\). This estimate is important later on in the uniqueness proof of Theorem 1.1. Recalling that \(({\mathbb {A}}(f))^*\in {\mathcal {L}}(L_2({\mathbb {S}}))\) is the \(L_2\)-adjoint of \({\mathbb {A}}(f)\in {\mathcal {L}}(L_2({\mathbb {S}}))\), we obtain for \({\overline{\omega }},\,\xi \in \widehat{L}_2({\mathbb {S}})\) that
meaning that the adjoint \(\big (\widehat{{\mathbb {A}}(f)}\big )^*:=\big ({\mathbb {A}}(f)|_{\widehat{L}_2({\mathbb {S}})}\big )^*\in {\mathcal {L}}(\widehat{L}_2({\mathbb {S}}))\) is given by \(\big (\widehat{{\mathbb {A}}(f)}\big )^*=P({\mathbb {A}}(f))^*.\)
Theorem 3.6
Let \(M>0\). Then, there exists a constant \(C=C(M)>0\) such that
for all \(\lambda \in {\mathbb {R}}\) with \(|\lambda |\ge 1,\)\(\xi \in \widehat{H}^1({\mathbb {S}}),\) and \(f\in H^2({\mathbb {S}})\) with \(\Vert f\Vert _{H^2}\le M.\)
In particular, \(\{\lambda \in {\mathbb {R}}\,:\,|\lambda |\ge 1\}\) is contained in the resolvent set of \(\big (\widehat{{\mathbb {A}}(f)}\big )^*\in {\mathcal {L}}(\widehat{H}^1({\mathbb {S}}))\) for each \(f\in H^2({\mathbb {S}})\).
Proof
Let \(M>0\). Taking advantage of the fact that \(\lambda -\big (\widehat{{\mathbb {A}}(f)}\big )^*\) is the \({\mathcal {L}}(\widehat{L}_2({\mathbb {S}}))\)-adjoint of \(\lambda -{\mathbb {A}}(f)\) for each \(\lambda \in {\mathbb {R}}\) and \(f\in H^r({\mathbb {R}})\), \(r>3/2\), it follows from (3.19) there exists a constant \(C=C(M)\) such that
for all \(\lambda \in {\mathbb {R}}\) with \(|\lambda |\ge 1,\)\(\xi \in \widehat{H}^1({\mathbb {S}}),\) and \(f\in H^2({\mathbb {S}})\) with \(\Vert f\Vert _{H^2}\le M.\) In order to show that \(\big (\widehat{{\mathbb {A}}(f)}\big )^*[\xi ]\in \widehat{H}^1({\mathbb {S}})\), we note that
where \({\mathbb {B}}_1(f)\) and \({\mathbb {B}}_2(f)\) are introduced in the proof of Lemma 3.1 and where
The arguments used to derive (3.12) show that \({\mathbb {A}}_{3,*}(f)[\xi ]\in H^1({\mathbb {S}})\) with
and together with (3.9), (3.10) we conclude that indeed \(\big (\widehat{{\mathbb {A}}(f)}\big )^*[\xi ]\in \widehat{H}^1({\mathbb {S}})\). Proceeding as in Theorem 3.5, we may write
with
satisfying
for any fixed \(\tau \in (1/2,1)\) and with a constant \(C=C(M)\). Moreover, since \({\mathbb {A}}(f)[1]\in H^1({\mathbb {S}})\), it follows that
again with \(C=C(M)\). The desired claim (3.31) follows now from (3.32), (3.33), and (3.34) by arguing as in Theorem 3.5. \(\square \)
Finally, given \(f\in H^r({\mathbb {S}})\), \(r>3/2\), let \(({\mathbb {B}}(f))^*\in {\mathcal {L}}(L_2({\mathbb {S}}))\) denote the adjoint of \({\mathbb {B}}(f)\in {\mathcal {L}}(L_2({\mathbb {S}}))\). The next lemma is also used later on in the uniqueness proof of Theorem 1.1.
Lemma 3.7
Given \(M>0\), there exists a constant \(C=C(M)\) such that for all \(f\in H^2({\mathbb {S}})\) with \(\Vert f\Vert _{H^2}\le M\) it holds that \(({\mathbb {B}}(f))^*\in {\mathcal {L}}(H^1({\mathbb {S}}))\) and
Proof
Given \(f\in H^2({\mathbb {S}})\), it is not difficult to show that
The desired estimate follows now by arguing as in Lemma 3.1. \(\square \)
4 The Muskat Problem with Surface Tension Effects
In this section we study the Muskat problem in the case when surface tension effects are included, that is for \(\sigma >0\). The main goal of this section is to prove Theorem 1.1 which is postponed to the end of the section. As a first step we shall take advantage of the results established in the previous sections to reexpress the contour integral formulation (1.1) as an abstract evolution equation of the form
with an operator \([f\mapsto \Phi _\sigma (f)]: H^2({\mathbb {S}})\rightarrow {\mathcal {L}}(H^3({\mathbb {S}}), L_2({\mathbb {S}}))\) defined in (4.7). The quasilinear character of the contour integral equation for \(\sigma >0\)—which is not obvious because of the coupling in (1.1a)\(_2\)—is expressed in (4.1) by the fact that \(\Phi _\sigma \) is nonlinear with respect to the first variable \(f\in H^2({\mathbb {S}})\), but is linear with respect to the second variable \(f\in H^3({\mathbb {S}})\) which corresponds to the third spatial derivatives of the function \(f=f(t,x)\) in the curvature term in (1.1a)\(_2\). A central part of the analysis in this section is devoted to showing that (4.1) is a parabolic problem in the sense that \(\Phi _\sigma (f)\)—viewed as an unbounded operator on \(L_2({\mathbb {S}})\) with definition domain \(H^3({\mathbb {S}})\)—is, for each \(f\in H^2({\mathbb {S}})\), the generator of a strongly continuous and analytic semigroup in \({\mathcal {L}}(L_2({\mathbb {S}}))\), which we denote by writing
This property needs to be verified before applying the abstract quasilinear parabolic theory outlined in [1,2,3,4,5] (see also [50]) in the particular context of (4.1).
We begin by solving the Eq. (1.1a)\(_2\) for \({\overline{\omega }}\). We shall rely on the invertibility properties provided in Theorems 3.3 and 3.5 and the fact that the Atwood number satisfies \(|a_\mu |<1\). In order to disclose the quasilinear structure of the Muskat problem with surface tension we address at this point the solvability of the equation
which for \(h=f\) coincides, up to a factor of 2, with (1.1a)\(_2\). The quasilinearity of the curvature term is essential here. For the sake of brevity we introduce
Since the values of \(\sigma >0\) and \(b_\mu >0\) are not important in the proof of Theorem 1.1 we set in this section
The solvability result in Proposition 4.1 (a) below is the main step towards writing (1.1) in the form (4.1). The decomposition of the solution operator provided at Proposition 4.1 (b) is essential later on in the proof of the generator property, as it enables us to use integration by parts when estimating some terms of leading order.
Proposition 4.1
-
(a)
Given \(f\in H^2({\mathbb {S}})\) and \(h\in H^3({\mathbb {S}})\), the function
$$\begin{aligned} {\overline{\omega }}(f)[h]:=(1+a_\mu {\mathbb {A}}(f))^{-1}\Big [\frac{h'''}{(1+f'^2)^{3/2}} -3\frac{f'f''h''}{(1+f'^2)^{3/2}}-\Theta h'\Big ] \end{aligned}$$is the unique solution to (4.3) in \(\widehat{L}_2({\mathbb {S}})\) and
$$\begin{aligned} {\overline{\omega }}\in \mathrm{C}^\omega (H^2({\mathbb {S}}), {\mathcal {L}}(H^3( {\mathbb {S}}), \widehat{L}_2({\mathbb {S}}))). \end{aligned}$$(4.5) -
(b)
Given \(f\in H^2({\mathbb {S}}) \) and \(h\in H^3({\mathbb {S}})\), let
$$\begin{aligned} {\overline{\omega }}_1(f)[h]&:=(1+a_\mu {\mathbb {A}}(f))^{-1} \Big [\frac{h''}{(1+f'^2)^{3/2}}-\Big \langle \frac{h''}{(1+f'^2)^{3/2}}\Big \rangle \Big ],\\ {\overline{\omega }}_2(f)[h]&:=(1+a_\mu {\mathbb {A}}(f))^{-1}\big [-\Theta h'+a_\mu T_{\mathrm{lot}}^A(f)[{\overline{\omega }}_1(f)[h]]\big ], \end{aligned}$$where \(T_{\mathrm{lot}}^A \) is defined in (3.28). Then:
-
(i)
\({\overline{\omega }}_1\in \mathrm{C}^\omega (H^2({\mathbb {S}}), {\mathcal {L}}(H^3({\mathbb {S}}),\widehat{H}^1({\mathbb {S}})))\) and \({\overline{\omega }}_2\in \mathrm{C}^\omega (H^2({\mathbb {S}}), {\mathcal {L}}(H^3({\mathbb {S}}), \widehat{L}_2({\mathbb {S}})));\)
-
(ii)
$$\begin{aligned} {\overline{\omega }}(f)=\frac{\mathrm{d}}{\mathrm{d}x}\circ {\overline{\omega }}_1(f)+{\overline{\omega }}_2(f); \end{aligned}$$
-
(iii)
Given \(\tau \in (1/2,1)\), there exists a constant C such that
$$\begin{aligned} \begin{aligned}&\Vert {\overline{\omega }}_1(f)[h]\Vert _2\le C\Vert h\Vert _{H^2}\\&\Vert {\overline{\omega }}_1(f)[h]\Vert _{H^{\tau }}+\Vert {\overline{\omega }}_2(f)[h]\Vert _2\le C\Vert h\Vert _{H^{2+\tau }} \end{aligned} \qquad \text {for all }h\in H^3({\mathbb {S}}). \end{aligned}$$(4.6)
-
(i)
Proof
Observing that the right hand side of (4.3) belongs to \(\widehat{L}_2({\mathbb {S}})\), the claim (a) follows from Theorems 3.3.
In order to prove (b) we first note that
and since by Theorem 3.5\([f\mapsto (1+a_\mu {\mathbb {A}}(f))^{-1}]\in \mathrm{C}^\omega (H^2({\mathbb {S}}),{\mathcal {L}}(\widehat{H}^1({\mathbb {S}}))),\) we conclude that \({\overline{\omega }}_1\) is well-defined together with \({\overline{\omega }}_1\in \mathrm{C}^\omega (H^2({\mathbb {S}}), {\mathcal {L}}(H^3({\mathbb {S}}), \widehat{H}^1({\mathbb {S}})))\). Recalling (3.27) and (3.28), it holds
This proves \({\overline{\omega }}_2\in \mathrm{C}^\omega (H^2({\mathbb {S}}), {\mathcal {L}}(H^3({\mathbb {S}}), \widehat{L}_2({\mathbb {S}}))) \) together with the claim (ii).
As for (iii), we note that the Theorems 3.3 and 3.5 imply that
and the estimate \(\Vert {\overline{\omega }}_1(f)[h]\Vert _{H^{\tau }}\le C\Vert h\Vert _{H^{2+\tau }}\), \(h\in H^3({\mathbb {S}})\), follows from the latter via interpolation. Finally, recalling Theorem 3.3 and (3.29), it holds
and the proof is complete. \(\square \)
Proposition 4.1 enables us to recast the contour integral formulation (1.1) of the Muskat problem with surface tension as the abstract quasilinear evolution problem (4.1), where
Proposition 4.1 and Lemma 3.1 imply that
In the following \(f\in H^2({\mathbb {S}})\) is kept fixed. In order to establish the generator property (4.2) for \(\Phi _\sigma (f)\) it is suitable to decompose this operator as the sum
where
The operator \(\Phi _{\sigma ,1}(f)\) can be viewed as the leading order part of \(\Phi _\sigma (f),\) while \(\Phi _{\sigma ,2}(f)\) is a lower order perturbation, see the proof of Theorem 4.3. We study first the leading order part \(\Phi _{\sigma ,1}(f).\) In order to establish (4.2) we follow a direct and self-contained approach pursued previously in [29, 33, 35] and generalized more recently in [32, 47,48,49] in the context of the Muskat problem. The proof of (4.2) uses a localization procedure which necessitates the introduction of certain partitions of unity for the unit circle.
To proceed, we choose for each integer \(p\ge 3\) a set \(\{\pi _j^p\,:\,{1\le j\le 2^{p+1}}\}\subset \mathrm{C}^\infty ({\mathbb {S}},[0,1])\), called p-partition of unity, such that
To each such p-partition of unity we associate a set \(\{\chi _j^p\,:\,{1\le j\le 2^{p+1}}\}\subset \mathrm{C}^\infty ({\mathbb {S}},[0,1])\) satisfying
As a further step we introduce the continuous path
which connects the operator \(\Phi _{\sigma ,1}(f)\) with the Fourier multiplier
where H denotes as usually the periodic Hilbert transform. Since H is the Fourier multiplier with symbol \((-i\mathrm{sign}(k))_{k\in {\mathbb {Z}}},\) it follows that \(\Phi _{\sigma ,1}(0)=- (\partial _x^4)^{3/4},\) that is the symbol of \(\Phi _{\sigma ,1}(0)\) is \((- |k|^3)_{k\in {\mathbb {Z}}}.\) In Theorem 4.2, which is the key argument in the proof of (4.2), we establish some commutator type estimates relating \(\Phi _{\sigma ,1}(\tau f)\) locally to some explicit Fourier multipliers. The proof of this result is quite technical and lengthy and uses to a large extent the outcome of Lemma A.1.
Theorem 4.2
Let \(f\in H^2({\mathbb {S}})\) and \(\mu >0\) be given. Then, there exist \(p\ge 3\), a p-partition of unity \(\{\pi _j^p\,:\, 1\le j\le 2^{p+1}\} \), a constant \(K=K(p)\), and for each \( j\in \{1,\ldots ,2^{p+1}\}\) and \(\tau \in [0,1]\) there exist operators
such that
for all \( j\in \{1,\ldots , 2^{p+1}\}\), \(\tau \in [0,1],\) and \(h\in H^3({\mathbb {S}})\). The operator \({\mathbb {A}}_{j,\tau }\) is defined by
where \(x_j^p\in I_j^p\) is arbitrary, but fixed.
Proof
Let \(p\ge 3\) be an integer which we fix later on in this proof and let \(\{\pi _j^p\,:\, 1\le j\le 2^{p+1}\}\) be a p-partition of unity, respectively, let \(\{\chi _j^p\,:\, 1\le j\le 2^{p+1}\} \) be a family associated to this p-partition of unity as described above. In the following, we denote by C constants which are independent of \(p\in {\mathbb {N}}\), \(h\in H^3({\mathbb {S}})\), \(\tau \in [0,1]\), and \(j \in \{1, \ldots , 2^{p+1}\}\), while the constants denoted by K may depend only on p.
Step 1: The lower order terms. Using the decomposition provided in the proof of Lemma 3.1 for the operator \({\mathbb {B}}\), we write
where, for the sake of brevity, we have set
Using integration by parts, we infer from (4.6) that
and we are left to consider the last two terms in (4.11).
Step 2: The first leading order term. Given \(1\le j\le 2^{p+1}\) and \(\tau \in [0,1]\), let
where \(x_j^p\in I_j^p\). In this step we show that if p is sufficiently large, then
for all \( j\in \{1,\ldots , 2^{p+1}\}\), \(\tau \in [0,1],\) and \(h\in H^3({\mathbb {S}})\). To this end we write
where
We first consider \(T_1[h].\) Recalling that \(\chi _j^p\pi _j^p=\pi _j^p\), algebraic manipulations lead us to
and the term \(T_{11}[h]\) may be expressed, after integrating by parts, as
Lemma A.1 (i) together with (4.6) yields
and
Hence, we need to estimate the term \(\Vert \pi _j^p{\overline{\omega }}_1'\Vert _2\) appropriately. The relation (3.27) and the definition of \({\overline{\omega }}_1 \) (see Proposition 4.1 (b)), yield
and the last term on the right hand side of (4.16) can be recast as
Integration by parts and Lemma A.1 (i) lead us to
Theorem 3.3, Lemma 3.2 (which can be applied as \((\pi _j^p{\overline{\omega }}_1)'\in \widehat{L}^2({\mathbb {S}})\)), (3.29) and (4.6) (both for \(\tau =3/4\)), and (4.16), (4.17) combined yield
and (4.6) now entails
Recalling that \(x_j^p\in I_j^p\subset J_j^p\) and \({{\,\mathrm{supp}\,}}\chi _j^p=\cup _{n\in {\mathbb {Z}}}(2\pi n+J_j^p ), \) the embedding \(H^{1}({\mathbb {S}})\hookrightarrow \mathrm{C}^{1/2}({\mathbb {S}})\) together with (4.14) (4.15), and (4.18) finally yield
provided that p is sufficiently large.
Noticing that
we write the term \(T_2[h]\) as
where
Though \(f_\tau '(x_j^p)\mathrm{id}_{\mathbb {R}}\) is not \( 2\pi \)-periodic, it is easy to see that the functions \(T_{2i}[h]\) still belong to \(L_2({\mathbb {S}}) \) for \(i\in \{1,2\}\). Since \(\chi _j^p\pi _j^p=\pi _j^p\), we have
where
Integrating by parts we obtain in view of (4.6) that
Since \(T_{21a}[h]\in L_2({\mathbb {S}}),\) it holds \(\Vert T_{21a}[h]\Vert _2=\Vert T_{21a}[h]\Vert _{L_2((-\pi ,\pi ))}\). Clearly, if \(x\in {{\,\mathrm{supp}\,}}(\mathbf{1}_{(-\pi ,\pi )}T_{21a}[h]),\) then
Letting \(J_j^p:=[a_j^p,b_j^p]\), \(p\ge 3\), \(1\le j\le 2^{p+1}\), we distinguish three cases.
-
(i)
If \(1\le j \le 2^{p}-1\), then \((-\pi ,\pi )\cap \big (2\pi n+ J_j^p\big )\ne \emptyset \) if and only if \(n=0\) and
$$\begin{aligned} (-\pi ,\pi )\cap J_j^p=[a_j^p,b_j^p]. \end{aligned}$$ -
(ii)
If \(2^p+3\le j\le 2^{p+1}\), then \((-\pi ,\pi )\cap \big (2\pi n+ J_j^p\big )\ne \emptyset \) if and only if \(n=-1\) and
$$\begin{aligned} (-\pi ,\pi )\cap (-2\pi + J_j^p) =[a_j^p-2\pi ,b_j^p-2\pi ]. \end{aligned}$$ -
(iii)
If \(j \in \{2^p, 2^{p}+1, 2^{p}+2\}\), then \((-\pi ,\pi )\cap \big (2\pi n+ J_j^p\big )\ne \emptyset \) if and only if \(n \in \{-1,0\}\), and
$$\begin{aligned} (-\pi ,\pi )\cap J_j^p =[a_j^p,\pi )\qquad \text {and}\qquad (-\pi ,\pi )\cap (-2\pi + J_j^p) =(-\pi ,-2\pi +b_j^p]. \end{aligned}$$
Assume that we are in the first case, that is \(1\le j\le 2^p-1.\) Let \(F_{\tau ,j}\) be the Lipschitz continuous function given by
Then \(\Vert F_{\tau ,j}'\Vert _\infty \le \Vert f'\Vert _\infty \). Taking into account that \(({{\,\mathrm{supp}\,}}\pi _j^p)\cap [a_j^p-\pi /2^p,b_j^p+\pi /2^p]\subset [a_j^p,b_j^p],\) it follows that
and, using integration by parts and (4.6), we arrive at
Moreover, combining Lemma A.1 (i) and (4.18), we find that
provided that p is sufficiently large. Altogether, we conclude that for \(1\le j\le 2^{p}-1\) it holds
Similar arguments apply also in the cases (ii) and (iii), and therefore the latter estimate actually holds for all \(1\le j\le 2^{p+1}\). Since \(T_{22}[h]\) can be estimated in the same way, we obtain that
provided that p is sufficiently large.
With regard to \(T_3[h],\) it holds
with
Integration by parts and Lemma A.1 (i) lead us to
A straight forward consequence of (4.16) is the following identity
Using once more the Hölder continuity of \(f'\), (3.29) and (4.6) (both with \(\tau =3/4\)) together with (4.17) yields that for p sufficiently large
We are left with the term
with \(T_{21}[h]\) defined above and with
Since
the estimate (4.18) and Lemma A.1 (i) for the first term, respectively integration by parts for the second term lead us, for p sufficiently large, to
Gathering (4.20) (which is valid also for \(C\Vert T_{21}[h]\Vert _2\) provided that we choose a larger p if required), (4.22), and (4.23), we conclude that
provided that p is sufficiently large. The estimate (4.13) follows now from (4.19), (4.21), and (4.24).
Step 2: The second leading order term. Given \(1\le j\le 2^{p+1}\) and \(\tau \in [0,1]\), let
where \(x_j^p\in I_j^p\). Similarly as in the previous step, it follows that
for all \( j\in \{1,\ldots , 2^{p+1}\}\), \(\tau \in [0,1],\) and \(h\in H^3({\mathbb {S}})\), provided that p is sufficiently large.
The desired claim (4.9) follows from (4.11), (4.12), (4.13), and (4.25). \(\square \)
We are now in a position to prove (4.2).
Theorem 4.3
Given \(f\in H^2({\mathbb {S}})\), it holds that
Proof
Let \(\Phi ^c_{\sigma }(f)=\Phi ^c_{\sigma ,1}(f)+\Phi ^c_{\sigma ,2}(f)\) denote the complexification of \(\Phi _\sigma (f)\) (the Sobolev spaces where \(\Phi ^c_{\sigma }(f)\) acts are now complex valued). In view of [46, Corollary 2.1.3] is suffices to show that \(-\Phi _\sigma ^c(f)\in {\mathcal {H}}(H^3({\mathbb {S}}), L_2({\mathbb {S}})).\) Moreover, for the choice \(\tau =3/4\) in Proposition 4.1 (b), we obtain together with Lemma 3.1, that \(\Phi _{\sigma ,2}^c(f)\in {\mathcal {L}}(H^{11/4}({\mathbb {S}}), L_2({\mathbb {S}})).\) Since \([L_2({\mathbb {S}}), H^3({\mathbb {S}})]_{11/12}=H^{11/4}({\mathbb {S}}),\) cf. (3.30), by [5, Theorem I.1.3.1 (ii)] we only need to show that
Recalling [5, Remark I.1.21 (a) ], we are left to find constants \(\omega >0\) and \(\kappa \ge 1\) such that
Let \(a>1\) be chosen such that
For each \(\alpha \in [a^{-1},a]\), let \({\mathbb {A}}_\alpha :H^3({\mathbb {S}})\rightarrow L_2({\mathbb {S}})\) denote operator \({\mathbb {A}}_\alpha :=-\alpha (\partial _x^4)^{3/4}.\) Then it is easy to see that for \(\kappa ':=1+a\) the following hold
Taking \(\mu :=1/(2\kappa ')\) in Theorem 4.2, we find \(p\ge 3\), a p-partition of unity \(\{\pi _j^p\,:\, 1\le j\le 2^{p+1}\} \), a constant \(K=K(p)\), and for each \(j\in \{1,\ldots ,2^{p+1}\}\) and \(\tau \in [0,1]\) operators \({\mathbb {A}}_{j,\tau }^c\in {\mathcal {L}}(H^3({\mathbb {S}}), L_2({\mathbb {S}}))\) (\({\mathbb {A}}_{j,\tau }^c\) is the complexification of \({\mathbb {A}}_{j,\tau }\) defined in (4.10)) such that
for all \( j\in \{1,\ldots , 2^{p+1}\}\), \(\tau \in [0,1],\) and \(h\in H^3({\mathbb {S}})\). We note that the relations (4.29) and (4.30) are both valid for \({\mathbb {A}}_{j,\tau }^c\) as \({\mathbb {A}}_{j,\tau }^c\in \{{\mathbb {A}}_\alpha \,:\,\alpha \in [a^{-1},a]\}\). It now follows from (4.30) and (4.31) that
for all \( j\in \{1,\ldots , 2^{p+1}\}\), \(\tau \in [0,1],\) and \(h\in H^3({\mathbb {S}})\). Since for each \(k\in {\mathbb {N}}\)
defines a norm equivalent to the standard \(H^k({\mathbb {S}})\)-norm, cf. [47, Remark 4.1], Young’s inequality together with (3.30) enables us to conclude from the previous inequality the existence of constants \(\omega >1\) and \(\kappa \ge 1\) with
Choosing \(\tau =1\) in (4.32) we obtain (4.28). Moreover, the estimate (4.32) for \(\lambda =\omega \), (4.29) (\(\Phi ^c_{\sigma ,1}(\tau f)={\mathbb {A}}_1\) for \(\tau =0\)), and the method of continuity [5, Proposition I.1.1.1] ensure that the property (4.27) also holds and the proof is complete. \(\square \)
We now come to the proof our first main result which uses on the one hand the abstract theory for quasilinear parabolic problems outlined in [1,2,3,4,5] (see also [50, Theorem 1.1]), and on the other hand a parameter trick which has been employed in various versions in [8, 34, 47,48,49, 57] in the context of improving the regularity of solutions to certain parabolic evolution equations. We point out that the parameter trick can only be used because the uniqueness claim of Theorem 1.1 holds in the setting of classical solution (the solutions in Theorem 1.1 possess though additional Hölder regularity properties, see the proof of Theorem 1.1).
Proof of Theorem 1.1
Let \({\mathbb {E}}_1:=H^3({\mathbb {S}})\), \({\mathbb {E}}_0:=L_2({\mathbb {S}})\), \(\beta :=2/3\) and \(\alpha :=r/3\). Then \({\mathbb {E}}_1\hookrightarrow {\mathbb {E}}_0\) is a compact embedding, \(0<\beta<\alpha <1\), and it follows from Theorem 4.3 and (4.8) that the abstract result [50, Theorem 1.1] may be applied in the context of the Muskat problem (4.1). Hence, given \(f_0\in H^r({\mathbb {S}})=[L_2({\mathbb {S}}), H^3({\mathbb {S}})]_\alpha \), (4.1) possesses a unique classical solution \(f=f(\,\cdot \,; f_0)\), that is
where \(T_+(f_0)\le \infty \), which has the property that
Concerning the uniqueness statement of Theorem 1.1 (i), it suffices to prove that if \(T>0\) and
solves (4.1) pointwise, then
cf. [50, Theorem 1.1]. Let thus f be a solution to (4.1) which satisfies (4.33). Since \(f\in C([0,T], H^r({\mathbb {S}}))\) and \(r>2\), we deduce from the Theorems 3.3 and 3.5 via interpolation that
Since \(\langle \kappa (f)\rangle =0\) and \(\sup _{t\in [0,T]}\Vert \kappa (f)\Vert _{H^{r-2}}\le C,\) it follows for \({\overline{\omega }}_1:={\overline{\omega }}_1(f)[f]=(1+a_\mu {\mathbb {A}}(f))^{-1}[\kappa (f)]\) (see Proposition 4.1) that
We next show that
It follows from the definitions of \(\Phi _{\sigma ,1}\) and \({\overline{\omega }}_1\) that
Using integration by parts, it is not difficult to derive, with the help of (4.35), the estimate
and we are left to consider the terms \(C_{0,1}(f)[{\overline{\omega }}_1']\) and \(f'C_{1,1}(f)[f,{\overline{\omega }}_1']\). Since \({\overline{\omega }}_1\in H^1({\mathbb {S}})\) for \(t\in (0,T],\) it is shown in Lemma 3.1 that \(C_{1,1}(f)[f,{\overline{\omega }}_1]\in H^1({\mathbb {S}})\) with
We estimate the terms on the right hand side of the latter identity in the \(H^{-1}\)-norm one by one. Given \(\varphi \in H^1({\mathbb {S}})\), integration by parts, (4.35), and Lemma A.1 (i) yield
and therewith
In order to estimate \(f'C_{1,1}(f)[f',{\overline{\omega }}_1]\) we write
where
Given \(\varphi \in H^1({\mathbb {S}})\), Fubini’s theorem yields for \(t\in (0,T]\)
and since \(|e^{i\xi }-1|\le C|\xi |\), respectively \(|e^{i\xi }-1|\le C|\xi |^{r-2}\), for all \(\xi \in {\mathbb {R}},\) the latter inequality together with (4.35) leads to
Arguing along the same lines we find for \(t\in (0,T]\), in view of \(|e^{i\xi }-2+e^{-i\xi }|\le C|\xi |^{r-1}\) for all \(\xi \in {\mathbb {R}},\) that
and therewith
Finally, the inequality \(|e^{i\xi }-2+e^{-i\xi }|\le C|\xi |^{2}\) for all \(\xi \in {\mathbb {R}}\) together with the Sobolev embedding \(H^{r-1}({\mathbb {S}})\hookrightarrow \mathrm{C}^{r-3/2}({\mathbb {S}})\) for \(r\ne 5/2,\) yield for \(t\in (0,T]\) that
hence
The latter estimate clearly holds also for \(r=5/2\). We have thus shown that
holds true. Similarly
Gathering (4.38)–(4.40), it follows that
Similarly, we get
and (4.37), (4.41), and (4.42) lead to
We now consider the second term \(\Phi _{\sigma ,2}.\) Given \(t\in (0,T]\), it holds
and Lemma 3.1 together with Theorem 3.3 yields
We now estimate \(\Vert {\mathbb {B}}(f)[{\overline{\omega }}_3]\Vert _{H^{-1}},\) where \({\overline{\omega }}_3:={\overline{\omega }}_3(f):=(1+a_\mu {\mathbb {A}}(f))^{-1}[T_{\mathrm{lot}}^A(f)[{\overline{\omega }}_1]]\in \widehat{L}_2({\mathbb {S}})\) for \(t\in (0,T]\). We begin by showing that the function \(T_{\mathrm{lot}}^A(f)[{\overline{\omega }}_1]\in \widehat{L}_2({\mathbb {S}})\), see (3.28), satisfies
Firstly we consider the difference \((f'{\mathbb {B}}_2(f)[{\overline{\omega }}_1])'-f'{\mathbb {B}}_2(f)[{\overline{\omega }}_1'],\) which we estimate, in view of (4.35) and Lemma 3.2, as follows
Secondly, it is not difficult to see that
We still need to estimate the terms of \(T_{\mathrm{lot}}^A(f)[{\overline{\omega }}_1]\) defined by means of the operators \(C_{n,m}\) introduced in Lemma A.1. This is done as follows
the last estimate following in a similar way as (4.39). Altogether, (4.45) holds true.
Given \(t\in (0,T]\), we compute for \(\varphi \in H^1({\mathbb {S}})\) that
where P is the orthogonal projection on \(\widehat{L}_2({\mathbb {S}}).\) This inequality together with Theorem 3.6 and (4.45) implies
Since for \(t\in (0,T] \) and \(\varphi \in H^1({\mathbb {S}})\)
Lemma 3.7 together with (4.46) lead us to
In view of (4.44) and (4.47) we conclude that
and the claim (4.36) follows from (4.43) and (4.48).
Recalling that \(f\in \mathrm{C}^1((0,T], L_2({\mathbb {S}}))\cap \mathrm{C}([0,T], H^r({\mathbb {S}})),\) (4.36) yields \(f\in \mathrm{BC}^1((0,T], H^{-1}({\mathbb {S}}))\) and the property (4.34) is now a straight forward consequence of (3.30). This proves the uniqueness claim in Theorem 1.1 and herewith the assertion (i). The claim (ii) follows directly from [50, Theorem 1.1], while the parabolic smoothing property stated at (iii) is obtain by using a parameter trick in the same way as in the proof of [49, Theorem 1.3]. The proof of Theorem 1.1 is now complete. \(\square \)
5 The Muskat Problem Without Surface Tension Effects
We now investigate the evolution problem (1.1) in the absence of the surface tension effects, that is for \(\sigma =0\). One of the main features of the Muskat problem with surface tension, namely the quasilinear character, seems to be lost as the curvature term disappears from the equations. Nevertheless, we show below that (1.1) can be recast as a fully nonlinear and nonlocal evolution problem
with \([f\mapsto \Phi (f)]\in C^\omega ( H^2({\mathbb {S}}), H^1({\mathbb {S}}))\) defined in (5.5). While the Muskat problem with surface tension is parabolic regardless of the initial data that are considered, in the case when \(\sigma =0\) we can prove that the Fréchet derivative \(\partial \Phi (f_0)\) generates a strongly continuous and analytic semigroup in \({\mathcal {L}}(H^1({\mathbb {S}}))\), more precisely that
only when requiring that the initial data \(f_0\in H^2({\mathbb {S}})\) are chosen such that the Rayleigh–Taylor condition is satisfied. Establishing (5.2) is the first goal of this section and this necessitates some preparations.
To begin, we solve the Eq. (1.1a)\(_2\), which is, up to a factor of 2, equivalent to
where
It is worth mentioning that in order to solve (5.3) for \({\overline{\omega }}\) in \(\widehat{H}^1({\mathbb {S}})\) it is required in Theorem 3.5 that the left hand side belongs to \(\widehat{H}^1({\mathbb {S}}),\) that is \(f\in H^2({\mathbb {S}})\), and this is precisely the regularity required also for the function in the argument of \({\mathbb {A}}\). Hence, (5.3) is no longer quasilinear, unless \(a_\mu =0\), see [47].
Proposition 5.1
Given \(f\in H^2({\mathbb {S}})\), there exists a unique solution \({\overline{\omega }}:={\overline{\omega }}(f)\in \widehat{H}^1({\mathbb {S}})\) to (5.3) and
Proof
Theorem 3.5 implies that
is the unique solution to (5.3) in \(\widehat{H}^1({\mathbb {S}}),\) and the regularity property (5.4) follows from Lemma 3.2. \(\square \)
In view of Proposition 5.1, (1.1) is equivalent to the Eq. (5.1), where \(\Phi :H^2({\mathbb {S}})\rightarrow \widehat{H}^1({\mathbb {S}})\) is given by
and it satisfies
cf. (3.6) and (5.4). With respect to our goal of proving Theorem 1.5, the fact that \(\Phi \) maps in \(\widehat{H}^1({\mathbb {S}})\) is not relevant, and therefore we shall not rely in this part on this property, but consider instead \(\Phi \) as a mapping in \(H^1({\mathbb {S}})\). In view of Lemma 2.2 and Proposition 5.1 the Rayleigh–Taylor condition (1.3) can be reformulated as
Indeed, recalling (2.1a)\(_2\) and (2.2)\(_2\), we obtain that
with \(V_\pm \) as defined in (2.5). The relation (5.7) follows now from (1.3), (5.3), and (5.5).
Since \(\Phi (f_0)\in \widehat{H}^1({\mathbb {S}}),\) it follows that (5.7) can hold only if \(\Theta >0\). We also note that (5.6) ensures that the set \({\mathcal {O}}\) of all initial data that satisfy the Rayleigh–Taylor condition (5.7), that is
is an open subset of \(H^2({\mathbb {S}})\) which is nonempty as it contains for example all constant functions.
In the following we fix an arbitrary \(f_0\in {\mathcal {O}}\) and prove the generator property (5.2) for the operator
where
is defined in Proposition 5.1. In view of (5.3) and of Proposition 5.1, we determine \(\partial {\overline{\omega }}(f_0)[f]\) as the solution to the equation
where, combining the Lemmas 3.2 and A.1 (i), we get
Establishing (5.2) is now more difficult than for the Muskat problem with surface tension, because there are several leading order terms to be considered when dealing with \(\partial \Phi (f_0)\), see the proof of Theorem 5.2. Besides, the Rayleigh–Taylor condition (5.7) does not appear in a natural way in the analysis and it has to be artificially built in instead. Indeed, let us first conclude from the Lemmas 3.1 and A.1 that
and let
denote the continuous path defined by
where
The function defined in (5.12) is related to \(\partial {\overline{\omega }}(f_0)[f]\). We emphasize that the last term on the right hand side of (5.12) has been introduced artificially with the purpose of identifying the function \( a_{\tiny {\text {RT}}}\) when setting \(\tau =0\), but also when relating \(\Psi (\tau )\) locally to certain Fourier multipliers, see Theorem 5.2 below. If \(\tau =1\), it follows that \(\Psi (1)=\partial \Phi (f_0),\) while for \(\tau =0\) we get
where we used once more the relation \({\mathbb {B}}(0)=H.\) We note that, since \(a_{\tiny {\text {RT}}}\) is in general not constant, the operator \(\Psi (0)\) is in general not a Fourier multiplier. However, we may benefit from the simpler structure of \(\Psi (0)\), compared to that of \(\partial \Phi (f_0)\), and the fact that the Rayleigh–Taylor condition holds to show that large real numbers belong to the spectrum of \(\Psi (0)\), see Proposition 5.3.
We now derive some estimates for the operator \(w\in C([0,1], {\mathcal {L}}(H^2({\mathbb {S}}), \widehat{H}^1({\mathbb {S}})))\), which are needed later on in the analysis. Let therefore \(\tau '\in (1/2,1)\). Since \(\Phi (f_0)\in H^1({\mathbb {S}})\), it follows from Theorem 3.3 and (3.15) (with \(r=1+\tau '\)) there exists a constant \(C>0\) such that
for all \(f\in H^2({\mathbb {S}})\) and \(\tau \in [0,1].\) Furthermore, Theorem 3.5 and (3.15) show that additionally
Using the interpolation property (3.30), we conclude from (5.14), (5.15) that
for all \(f\in H^2({\mathbb {S}})\) and \(\tau \in [0,1].\)
The following result is the main step towards proving the generator property (5.2). Below \((-\partial _x^2)^{1/2} \) stands for the Fourier multiplier with symbol \((|k|)_{k\in {\mathbb {Z}}}\), and the following identity is used
Theorem 5.2
Let \(f_0\in H^2({\mathbb {S}})\) and \(\mu >0\) be given. Then, there exist \(p\ge 3\), a p-partition of unity \(\{\pi _j^p\,:\, 1\le j\le 2^{p+1}\} \), a constant \(K=K(p)\), and for each \( j\in \{1,\ldots ,2^{p+1}\}\) and \(\tau \in [0,1]\) there exist operators
such that
for all \( j\in \{1,\ldots , 2^{p+1}\}\), \(\tau \in [0,1],\) and \(f\in H^2({\mathbb {S}})\). The operator \({\mathbb {A}}_{j,\tau }\) is defined by
where \(x_j^p\in I_j^p\) is arbitrary, but fixed, and where
Proof
Let \(p\ge 3\) be an integer which we fix later on in this proof and let \(\{\pi _j^p\,:\, 1\le j\le 2^{p+1}\}\) be a p-partition of unity, respectively let \(\{\chi _j^p\,:\, 1\le j\le 2^{p+1}\} \) be a family associated to this partition. We denote by C constants which are independent of \(p\in {\mathbb {N}}\), \(f\in H^2({\mathbb {S}})\), \(\tau \in [0,1]\), and \(j \in \{1, \ldots , 2^{p+1}\}\), while the constants denoted by K may depend only upon p.
The lower order terms. We first note that
The relations (3.6) (with \(r=7/4\)) and (5.14) (with \(\tau '=3/4\)) yield
and since \(\max _{ \tau \in [0,1]}(\Vert \alpha _\tau \Vert _{H^1}+\Vert \beta _\tau \Vert _{H^1})\le C,\) it also holds that
Therewith we get
Moreover, combining (5.11), (3.10) (with \(r=7/4\) and \(\tau =3/4\)), Lemma A.1 (ii) (with \(\tau =3/4\) and \(r=15/8\)), and (5.16) (with \(\tau '=3/4\)), we may write
where
Consequently, we are left to estimate the \(L_2\)-norm of the difference
Higher order terms I. Given \(1\le j\le 2^{p+1}, \) we set
Since \({\mathbb {B}}_1(f_0)[{\overline{\omega }}_0],\, C_{1,1}(f_0)[f_0,{\overline{\omega }}_0]\in H^1({\mathbb {S}})\hookrightarrow \mathrm{C}^{1/2}({\mathbb {S}})\) and \(\chi _j^p\pi _j^p=\pi _j^p\), it follows that
provided that p is sufficiently large.
Higher order terms II. Letting
it holds that
where
The first term may be estimated, by using integration by parts, in a similar way as the term \(T_{11}[h]\) in the proof of Theorem 4.2, that is
Besides, the same arguments used to derive (4.21) show that for p sufficiently large
Finally, it holds that
and, recalling that \(\chi _j^p=1\) on \({{\,\mathrm{supp}\,}}\pi _j^p\), we obtain, by using integration by parts, Lemma A.1 (i), and the fact that \({\overline{\omega }}_0\in H^1({\mathbb {S}})\hookrightarrow \mathrm{C}^{1/2}({\mathbb {S}})\) the estimate
provided p is sufficiently large. Summarizing, we have shown that
and similarly we get
Higher order terms III. We are left to consider the function
where, for the sake of brevity, we have set
Let further
We first derive an estimate for the \(L_2\)-norm of \(\pi _j^p w'.\) To this end we differentiate (5.12) once to obtain, in view of (3.27), (5.10), and Lemma A.1 (i)–(ii), that
Combining (3.10) (with \(\tau =3/4\) and \(r=7/4\)), (3.29) (with \(\tau =3/4\)), (4.17), (5.14) and (5.16) (both with \(\tau '=3/4\)), and Lemma A.1 (i)–(ii) (with \(\tau =3/4\) and \(r=15/8\)) we get that
The relation (5.23) together with Theorem 3.3, Lemma A.1 (i), and (5.14) (with \(\tau '=3/4\)) now yields
We now consider the second term on the right hand side of (5.22). Letting
we write
where
The arguments that led to (4.19) together with (5.25) show that
provided that p is sufficiently large, while arguing as in the derivation of (4.21) we obtain that
Concerning \(T_6[f]\), we find, by using fact that the Hilbert transform satisfies \(H^2=-\mathrm{id}_{L_2({\mathbb {S}})}\), the following relation
and, since integration by parts and (5.14) (with \(\tau '=3/4\)) yield
we conclude that
Combining (3.16) and (5.23), we further get
and the estimates (4.17), (5.14) (with \(\tau '=3/4\)), (5.24), together with the arguments used to estimate \(\Vert T_2[f]\Vert _2\) show, for p sufficiently large, that
Altogether, we have shown that
Letting
we obtain in a similar way, that
provided that p is sufficiently large, and therewith we conclude that
Final step. Using the identity \({\overline{\omega }}_0=-c_\Theta f_0'-a_\mu {\mathbb {A}}(f_0)[{\overline{\omega }}_0],\) it is not difficult to see that
and (5.19), (5.20), (5.21), and (5.26) immediately yield (5.17). \(\square \)
Making use of the fact that for \(f_0\in {\mathcal {O}}\) the Rayleigh–Taylor condition \(a_{\tiny {\text {RT}}}>0\) is satisfied, it follows from the general result in Proposition 5.3 below that \(\Psi (0)\) contains in its resolvent set all sufficiently large real numbers.
Proposition 5.3
Let \(a\in H^1({\mathbb {S}})\) be a positive function. Then, there exists \(\omega _0\ge 1\) with the property that \(\lambda +H[a\partial _x]\in \mathrm{Isom} (H^2({\mathbb {S}}), H^1({\mathbb {S}}))\) for all \(\lambda \in [\omega _0,\infty )\).
Proof
Let \(m:=\min _{{\mathbb {S}}} a>0\). We introduce the continuous path \([\tau \mapsto B(\tau )]:[0,1]\rightarrow {\mathcal {L}}(H^2({\mathbb {S}}), H^1({\mathbb {S}}))\) via
Since \(\lambda +B(0)\) is the Fourier multiplier with symbol \((\lambda +m |k|)_{k\in {\mathbb {Z}}}\), it is obvious that \(\lambda +B(0)\) is invertible for all \(\lambda >0\). If \(\lambda \) is sufficiently large, we show below that \(\lambda +B(1)=\lambda +H[a\partial _x]\) has this property too. To this end we prove that for each \(\mu >0\) there exists \(p\ge 3\), a p-partition of unity \(\{\pi _j^p\,:\, 1\le j\le 2^{p+1}\}\), a constant \(K=K(p)\), and for each \( j\in \{1,\ldots ,2^{p+1}\}\) and \(\tau \in [0,1]\) there exist operators
such that
for all \( j\in \{1,\ldots , 2^{p+1}\}\), \(\tau \in [0,1],\) and \(f\in H^2({\mathbb {S}})\). The operators \({\mathbb {B}}_{j,\tau }\) are the Fourier multipliers
with \(x_j^p\in I_j^p\). Indeed, given \(p\ge 3\), let \(\{\pi _j^p\,:\, 1\le j\le 2^{p+1}\}\) be a p-partition of unity and let \(\{\chi _j^p\,:\, 1\le j\le 2^{p+1}\} \) be a family associated to this partition. Integrating by parts we get
provided that p is sufficiently large, and (5.27) follows.
A simple computation shows that there exists \(\kappa \ge 1\) such that
for all \(f\in H^2({\mathbb {S}})\), \(\alpha \ge m\), and \(\lambda \in [1,\infty )\). Set \(\mu :=1/2\kappa \) in (5.27). Since \(a_\tau \ge m\), it follows from (5.27) and (5.28) that
for all \(f\in H^2({\mathbb {S}})\), \(\lambda \ge 1\), \(\tau \in [0,1]\), and \( j\in \{1,\ldots , 2^{p+1}\}\). The arguments at the very and of the proof of Theorem 4.3 enable us to conclude the existence of constants \(\beta \in (0,1)\) and \(\omega _0\ge 1\) with
for all \(f\in H^2({\mathbb {S}})\), \(\lambda \ge \omega _0\), and \(\tau \in [0,1]\). The continuity method [5, Proposition I.1.1.1] and the previous observation that \(\lambda +B(0)\in \mathrm{Isom}(H^2({\mathbb {S}}), H^1({\mathbb {S}}))\) for \(\lambda >0\) yield the desired conclusion. \(\square \)
We are now in a position to derive the desired generator property (5.2).
Theorem 5.4
Given \(f_0\in {\mathcal {O}}\), it holds that
Proof
Given \(f_0\in {\mathcal {O}}\) and \(\tau \in [0,1],\) let \(\alpha _\tau \) and \(\beta _\tau \) denote the functions introduced in Theorem 5.2. The Rayleigh–Taylor condition \(a_{\tiny {\text {RT}}}>0\) ensures there exists a constant \(\eta \in (0,1)\) such that
for all \(\tau \in [0,1]\). Given \(\alpha \in [\eta ,1/\eta ]\) and \(|\beta |<1/\eta \), let \({\mathbb {A}}_{\alpha ,\beta }\) denote the Fourier multiplier
It is not difficult to prove there exists \(\kappa _0\ge 1 \) such that the complexification of \({\mathbb {A}}_{\alpha ,\beta }\) (denoted again by \({\mathbb {A}}_{\alpha ,\beta }\)) satisfies
for all \(\alpha \in [\eta ,1/\eta ]\), \(|\beta |<1/\eta \), \(\mathrm{Re}\lambda \ge 1\), and \(f\in H^2({\mathbb {S}})\). Observing that the operators \({\mathbb {A}}_{j,\tau }\) found in Theorem 5.2 belong to the family \(\{{\mathbb {A}}_{\alpha ,\beta }\,:\,{\alpha \in [\eta ,1/\eta ], |\beta |<1/\eta }\} \) and that
for all \(\lambda \in {\mathbb {R}}\) which are sufficiently large, cf. Proposition 5.3, the arguments in the proof of Theorem 4.3 together with (5.29) and Theorem 5.2 lead us to the desired claim. \(\square \)
We conclude this section with the proof of Theorem 1.5.
Proof of Theorem 1.5
The proof follows by using the fully nonlinear parabolic theory in [46, Chapter 8], (5.6), and Theorem 5.4. The details of proof are identical to those in the nonperiodic case, cf. [48, Theorem 1.2], and therefore we omit them. \(\square \)
6 Stability Analysis
In this section we identify the equilibria of the Muskat problem (1.1) and study their stability properties.
The Muskat problem without surface tension We first infer from Remark 3.4 that \(f\in H^2({\mathbb {S}})\) is a stationary solution to (1.1) (with \(\sigma =0\)) if and only if f is constant also with respect to x. Besides, as pointed out in Sect. 5, if f is a solution to (1.1) as found in Theorem 1.5, then \(\Phi (f(t))\in \widehat{H}^1({\mathbb {S}})\) for all t in the existence interval of f, hence the mean integral of the initial datum is preserved by the flow. Recalling also the invariance property (1.2), we shall only address the stability issue for the 0 equilibrium under perturbed initial data with zero integral mean. Hence, we are led to consider the evolution problem
where
is the restriction of the operator defined in (5.5). Recalling (5.8), it follows from the relations \({\overline{\omega }}(0)=0\), \({\mathbb {A}}(0)=0\), and \({\mathbb {B}}(0)=H\), that
which identifies the spectrum \(\sigma (\partial \Phi (0))\) as being the set
Moreover, it is easy to verify that this Fourier multiplier is the generator of a strongly continuous and analytic semigroup in \({\mathcal {L}}(\widehat{H}^1({\mathbb {S}})).\) This enable us to use the fully nonlinear principle of linearized stability, cf. [46, Theorem 9.1.1], and prove in this way the exponential stability of the zero solution.
Proof of Theorem 1.6
The claim follows from (6.2), the property \(-\partial \Phi (0)\in {\mathcal {H}}(\widehat{H}^2({\mathbb {S}}),\widehat{H}^1({\mathbb {S}}))\), and the fact that \(\mathrm{Re}\lambda \le -c_\Theta \) for all \(\lambda \in \sigma (\partial \Phi (0))\) via [46, Theorem 9.1.1]. \(\square \)
The Muskat Problem with Surface Tension For \(\sigma >0\) the stability analysis is more intricate. Before presenting the complete picture of the equilibria we notice that also in this case the mean value of the initial data is preserved by the flow. This aspect and the invariance property (1.2) enable us to restrict our stability analysis to the setting of solutions with zero integral mean.
In view of Remark 3.4, a function \(f\in \widehat{H}^3({\mathbb {S}})\) is a stationary solution to (1.1) if and only if it solves the capillarity equation
This equation has been discussed in detail in [28]. If \(\lambda \le 0\), the Eq. (6.3) has by the elliptic maximum principle a unique solution in \(\widehat{H}^3({\mathbb {S}})\), the trivial equilibrium \(f=0\). However, if \(\lambda >0\), there may exist also finger-shaped solutions to (6.3), see Fig. 1, which are all symmetric with respect to the horizontal lines through the extrema but also with respect to the points where they intersect the x-axis. In particular, each equilibrium in \(\widehat{H}^3({\mathbb {S}})\) is the horizontal translation of an even equilibrium. We now view \(\lambda >0\) as a bifurcation parameter in the Eq. (6.3) and we shall refer to \((\lambda ,f)\) as being the solution to (6.3). The following theorem provides a complete description of the set of even equilibria to the Muskat problem with surface tension (and in virtue of (1.2) also of the set of all equilibria).
Theorem 6.1
Let
where B is the beta function. The even solutions to (6.3) are organized as follows.
-
(a)
If \(\lambda \le \lambda _*\),Footnote 5 then (6.3) has only the trivial solution.
-
(b)
Let \(\lambda >\lambda _* \).
-
(i)
The Eq. (6.3) has even solutions of minimal period \(2\pi \) if and only if \(\lambda _*<\lambda <1\). More precisely, for each \(\lambda \in (\lambda _*,1)\), (6.3) has exactly two even solutions \((\lambda ,\pm f_\lambda )\) of minimal period \(2\pi \). These solutions are real-analytic, \(|f_{\lambda _1}|\le |f_{\lambda _2}|\) for \(\lambda _2<\lambda _1\), \(\Vert f_\lambda \Vert _\infty \rightarrow 0\) for \(\lambda \nearrow 1\), and
$$\begin{aligned} \Vert f_\lambda \Vert _\infty =|f_\lambda (0)|\nearrow \sqrt{2/\lambda _*},\quad \Vert f_\lambda '\Vert _\infty =|f_\lambda '(\pi /2)|\nearrow \infty \quad \text { for }\lambda \searrow \lambda _*. \end{aligned}$$ -
(ii)
The Eq. (6.3) has even solutions of minimal period \(2\pi /\ell \), \(2\le \ell \in {\mathbb {N}}\), if and only if \(\ell ^{2}\lambda _*<\lambda <\ell ^{2}\). More precisely, for each \(\lambda \in (\ell ^{2}\lambda _*,\ell ^{2}),\) (6.3) has exactly two even solutions \((\lambda ,\pm f_\lambda )\) of minimal period \(2\pi /\ell \) and
$$\begin{aligned} f_\lambda =\ell ^{-1}f_{\lambda \ell ^{-2}}(\ell \, \cdot \, ) \end{aligned}$$where \(f_{\lambda \ell ^{-2}}(\ell \, \cdot \, )\) is the function identified at (ii).
-
(i)
-
(c)
If we consider (6.3) as an abstract bifurcation problem in \({\mathbb {R}}\times \widehat{H}^3_e({\mathbb {S}}),\) where
$$\begin{aligned} \widehat{H}^3_e({\mathbb {S}}):=\{f\in H^3_e({\mathbb {S}})\,:\, f\text { is even}\}, \end{aligned}$$then the global bifurcation curve arising from \((\ell ^2,0),\)\(1\le \ell \in {\mathbb {N}},\) and described at (b), admits in a neighborhood of \((\ell ^2,0)\) a real-analytic parametrization
$$\begin{aligned} (\lambda _\ell ,f_\ell ):(-\varepsilon _\ell ,\varepsilon _\ell )\rightarrow (0,\infty )\times \widehat{H}^3_e({\mathbb {S}}) \end{aligned}$$such that
$$\begin{aligned} \left\{ \begin{array}{lll} \lambda _\ell (s)=\ell ^2-\frac{3\ell ^4}{8}s^2+O(s^4) \quad \text {in }{\mathbb {R}},\\ f_\ell (s)= s\cos (\ell x)+O(s^2) \qquad \text {in } \widehat{H}^3_e({\mathbb {S}}) \end{array} \right. \qquad \text {for }s\rightarrow 0. \end{aligned}$$
Proof
The claims (a) and (b) are established in [28]. The last claim follows by applying the theorem on bifurcations from simple eigenvalues due to Crandall and Rabinowitz, cf. [19]. The details are similar to those in the proof of [31, Theorem 6.1]. \(\square \)
With respect to Theorem 6.1 we add the following remark.
Remark 6.2
-
(i)
Because \(\lambda _*\approx 0.3\), for certain \(\lambda \in (\ell ^2\lambda _*,\ell ^2)\) with \(\ell \ge 2\) there exist nontrivial solutions to (6.3) with minimal period different than \(2\pi /\ell \), see Fig. 1.
-
(ii)
As pointed out in [28], these finger-shaped equilibria are in correspondence to certain solutions to the mathematical pendulum equation
$$\begin{aligned} \theta ''+\lambda \sin \theta =0. \end{aligned}$$ -
(iii)
The global bifurcation curves may be continued beyond \(\lambda _*\ell ^2\), but outside the setting of interfaces parametrized as graphs.
-
(iv)
Because \(\lambda _\ell '(0)=0>\lambda _\ell ''(0)\), we may assume that \(s\lambda _\ell '(s)<0\) for all \(s\in (-\varepsilon _\ell ,\varepsilon _\ell ){\setminus }\{0\}\). This aspect is of relevance when studying the stability properties of the finger-shaped equilibria identified above.
In order to address the stability properties of the equilibria to (1.1), we first reformulate the problem by incorporating \(\lambda \) as a parameter. To this end we define \(\Phi :{\mathbb {R}}\times \widehat{H}^2({\mathbb {S}})\rightarrow {\mathcal {L}}(\widehat{H}^3({\mathbb {S}}),\widehat{L}_{2}({\mathbb {S}}))\) according to
where \(b_\mu \) is the constant introduced in (4.4). Then, it follows from the analysis in Sect. 4 that \(\Phi \in C^\omega ({\mathbb {R}}\times \widehat{H}^2({\mathbb {S}}),{\mathcal {L}}(\widehat{H}^3({\mathbb {S}}), \widehat{L}_{2}({\mathbb {S}}))),\) and the problem (1.1) is equivalent, for solutions with zero integral mean, to the quasilinear evolution problem
It is not difficult to see that the linearization \( \Phi (\lambda ,0)\in {\mathcal {L}}(\widehat{H}^3({\mathbb {S}}), \widehat{L}_{2}({\mathbb {S}}))\) is a Fourier multiplier with spectrum \(\sigma ( \Phi (\lambda ,0))\) that consists only of the eigenvalues \(\{-\sigma b_\mu (|k|^3-\lambda |k|)\,:\,k\in {\mathbb {Z}}{\setminus }\{0\}\}.\) Moreover, \( \Phi (\lambda ,0)\) generates a strongly continuous and analytic semigroup in \({\mathcal {L}}( \widehat{L}_{2}({\mathbb {S}})) \) for all \(\lambda \in {\mathbb {R}}\). We are now in a position to prove Theorem 1.3 where we exploit the quasilinear principle of linearized stability in [50, Theorem 1.3].
Proof of Theorem 1.3
We first address the stability of the zero solution \(f=0\) to (1.1). Assume first that \(\lambda <1\). In this case all eigenvalues of \(\Phi (\lambda ,0)\) are negative, more precisely \(\mathrm{Re}z\le -\sigma b_\mu (1-\lambda )<0\) for all \(z\in \sigma (\Phi (\lambda ,0))\). The quasilinear principle of linearized stability [50, Theorem 1.3] applied to (6.5) yields the first claim of Theorem 1.3.
In the second case when \(\lambda >1\), the intersection \( \sigma (\partial _f\Phi (\lambda ,0))\cap [\mathrm{Re}\lambda >0]\) consists of a finite number of positive eigenvalues and we may apply the instability result in [50, Theorem 1.4] to derive the assertion (ii) in Theorem 1.3.
In the remaining part we discuss the stability properties of small finger-shaped solutions. To this end we denote by \({\mathbb {A}}_\ell (s)\) the linearized operator
where \(\partial _f \Phi \in {\mathcal {L}}( \widehat{H}^2({\mathbb {S}}),{\mathcal {L}}(\widehat{H}^3({\mathbb {S}}), \widehat{L}_{2}({\mathbb {S}})))\) is the Fréchet derivative of the mapping \( \Phi \) with respect to the variable f. We point out that \({\mathbb {A}}_\ell (0)=\Phi (\ell ^2,0).\)
Let us first note that for \(\ell \ge 2\) the spectrum \( \sigma ({\mathbb {A}}_\ell (0))\) contains a finite number of positive eigenvalues (this number increases with \(\ell \)). Since a set consisting of finitely many eigenvalues of \({\mathbb {A}}_\ell (s)\) changes continuously with \(s\in (-\varepsilon _\ell ,\varepsilon _\ell )\), cf. [44, Chapter IV], we infer from [5, Theorem I.1.3.1 (i)] that \(-{\mathbb {A}}(s)\in {\mathcal {H}}(\widehat{H}^3({\mathbb {S}}),\widehat{L}_2({\mathbb {S}}))\) and that \(\sigma ({\mathbb {A}}(s))\) contains only finitely many eigenvalues with positive real part if \(\varepsilon _\ell \) is sufficiently small. Thus, we may appeal to [50, Theorem 1.4] to conclude that if \(\lambda =\lambda _\ell (s),\)\(0<|s|<\varepsilon _\ell ,\)\(\ell \ge 2\), then \(f_\ell (s)\) is an unstable equilibrium to (1.1).
The situation when \(\ell =1\) is special, because \(\sigma ({\mathbb {A}}_1(s))\) has for \(s=0\), except for the eigenvalue 0, only negative eigenvalues. We show below that when letting s vary in \((-\varepsilon _1,\varepsilon _1)\) the operator \({\mathbb {A}}_1(s)\), \(0<|s|<1\), has a positive eigenvalue z(s) which corresponds to the zero eigenvalue of \( {\mathbb {A}}_1(0)\). To this end we associate to a periodic function h the function \({\check{h}}\) defined by
Observing that and , \(f\in \widehat{H}^2({\mathbb {S}})\), \({\overline{\omega }}\in \widehat{L}_2({\mathbb {S}})\), and that
cf. Proposition 4.1, it follows that the operator \(\Phi \) introduced in (6.4) satisfies
Hence, letting \(\widehat{L}_{2,e}({\mathbb {S}}):=\{f\in \widehat{L}_2({\mathbb {S}})\,:\, \)f\( \text {is even}\}\) and \(\widehat{H}^r_e({\mathbb {S}}):=\widehat{H}^r({\mathbb {S}})\cap L_{2,e}({\mathbb {S}}),\)\(r\ge 0\), it follows that \(\Phi \in C^\omega ({\mathbb {R}}\times \widehat{H}^2_e({\mathbb {S}}), {\mathcal {L}}(\widehat{H}^3_e({\mathbb {S}}),\widehat{L}_{2,e}({\mathbb {S}}))),\) the linearization \( {\mathbb {A}}_1(0)\in {\mathcal {L}}(\widehat{H}^3_e({\mathbb {S}}), \widehat{L}_{2,e}({\mathbb {S}}))\) being the Fourier multiplier
Let \(\Psi :{\mathbb {R}}\times \widehat{H}^3_e({\mathbb {S}})\rightarrow \widehat{L}_{2,e}({\mathbb {S}})\) be the real-analytic mapping defined by \(\Psi (\lambda ,f):=\Phi (\lambda ,f)[f]\). Noticing that \(\partial _f\Psi (\lambda _1(s),f_1(s))={\mathbb {A}}_1(s)\), it follows that 0 is a simple eigenvalue of \(\partial _f\Psi (1,0)\) and \(\mathrm{Ker\,} \partial _f\Psi (1,0)=\mathrm{span}\{\cos (x)\}.\) Since additionally \(\partial _{\lambda f}\Psi (1,0)[\cos (x)]=\sigma b_\mu \cos (x)\not \in \mathrm{Im}\partial _f\Psi (1,0)\), the principle of exchange of stability, cf. [20, Theorem 1.16], together with Remark 6.2 (iv) implies that the zero eigenvalue of \(\partial _f\Psi (1,0)\) perturbs along the bifurcation curve through \((\lambda _1,f_1)\) into a positive eigenvalue z(s) of \({\mathbb {A}}_1(s)\), \(0<|s|<\varepsilon _1\), and moreover
Hence, if \(\varepsilon _1\) is sufficiently small, the operator \( {\mathbb {A}}_1(s)\), \(0<|s|<\varepsilon _1\), has a positive eigenvalue z(s). Moreover, \( {\mathbb {A}}_1(s)\) has at most two eigenvalues with positive real part. [50, Theorem 1.4] yields now that if \(\lambda =\lambda _1(s),\)\(0<|s|<\varepsilon _1\), then \(f_1(s)\) is an unstable equilibrium. \(\square \)
Notes
When \(\sigma =0\) we require that the Eq. (1.1a) are satisfied also at \(t=0\).
We write \(\dot{f}\) to denote the derivative \(\mathrm{d}f/\mathrm{d}t\).
Letting \([\,\cdot \,,\,\cdot \, ]_\theta \) denote complex interpolation functor, it is well-known that
$$\begin{aligned}{}[H^{s_0}({\mathbb {S}}),H^{s_1}({\mathbb {S}})]_\theta =H^{(1-\theta )s_0+\theta s_1}({\mathbb {S}}),\qquad \theta \in (0,1),\, -\infty< s_0\le s_1<\infty . \end{aligned}$$(3.30)A rough estimate for \(\lambda _*\) is \(\lambda _*\approx 0.3\).
- $$\begin{aligned} B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots ,b_n,{\overline{\omega }}](x):=\mathrm{PV}\int _{{\mathbb {R}}} \frac{{\overline{\omega }}(x-s)}{s}\frac{\prod _{i=1}^{n}\big (\delta _{[x,s]} b_i /s\big )}{\prod _{i=1}^{m}\big [1+\big (\delta _{[x,s]} a_i /s\big )^2\big ]} \mathrm{d}s \end{aligned}$$
are considered. The functions \(a_1,\ldots , a_{m},\, b_1, \ldots , b_n:{\mathbb {R}}\rightarrow {\mathbb {R}}\) are Lipschitz functions and \({\overline{\omega }}\in L_2({\mathbb {R}}).\) It is shown in [48, 49] that these operators extend to bounded multilinear operators on certain products of Sobolev spaces on \({\mathbb {R}}\).
Recall that \(\tau _s \) stands for the right translation. Moreover, \(\widehat{h}(k)\), \(k\in {\mathbb {Z}}\), is the k-th Fourier coefficient of \(h\in L_1({\mathbb {S}})\).
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Appendix A. Some Technical Results
Appendix A. Some Technical Results
In Lemma A.1 we establish the boundedness of a family of multilinear singular integral operators in certain settings that are motivated by the analysis in the previous sections. The nonperiodic counterparts of the estimates derived below have been obtained previously in [48, 49].Footnote 6
Lemma A.1
-
(i)
Given \(m,\, n\in {\mathbb {N}}\) and Lipschitz functions \(a_1,\ldots , a_{m},\, b_1, \ldots , b_n:{\mathbb {R}}\rightarrow {\mathbb {R}}\), the singular integral operator \(C_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots ,b_n,\,\cdot \,]\) defined by
$$\begin{aligned} C_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots ,b_n,{\overline{\omega }}](x):=\mathrm{PV}\int _{-\pi }^\pi \frac{{\overline{\omega }}(x-s)}{s} \frac{\prod _{i=1}^{n}\big (\delta _{[x,s]} b_i /s\big )}{\prod _{i=1}^{m}\big [1+\big (\delta _{[x,s]} a_i /s\big )^2\big ]} \mathrm{d}s \end{aligned}$$satisfies \(\Vert C_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots ,b_n,\,\cdot \,]\Vert _{{\mathcal {L}}(L_2({\mathbb {S}}),L_2((-\pi ,\pi )))}\le C\prod _{i=1}^{n} \Vert b_i'\Vert _{\infty },\) with a constant C that depends only on \(n,\, m\) and \(\max _{i=1,\ldots , m}\Vert a_i'\Vert _{\infty }.\)
In particular, \(C_{n,m}\in \mathrm{C}^{1-}((W^1_\infty ({\mathbb {S}}))^{m}, {\mathcal {L}}_{n+1}( (W^1_\infty ({\mathbb {S}}))^{n}\times L_2({\mathbb {S}}), L_2({\mathbb {S}}))).\)
-
(ii)
Let \(m\in {\mathbb {N}}\), \(1\le n\in {\mathbb {N}}\), \(r\in (3/2,2)\), and \(\tau \in (5/2-r,1).\) Then:
-
(ii1)
Given \(a_1,\ldots , a_m\in H^r({\mathbb {S}})\) and \( b_1,\ldots , b_n,\, {\overline{\omega }}\in \mathrm{C}^\infty ({\mathbb {S}})\), there exists a constant C that depends only on n, m, r, \(\tau \), and \(\max _{i=1,\ldots , m}\Vert a_i\Vert _{H^r({\mathbb {S}})}\) such that
$$\begin{aligned}&\Vert C_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,{\overline{\omega }}]\Vert _{L_2({\mathbb {S}})}\le C\Vert {\overline{\omega }}\Vert _{H^\tau ({\mathbb {S}})}\Vert b_1\Vert _{H^1({\mathbb {S}})}\prod _{i=2}^n\Vert b_i\Vert _{H^r({\mathbb {S}})} \end{aligned}$$(A.1)and
$$\begin{aligned} \begin{aligned}&\Vert C_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n, {\overline{\omega }}]-C_{n-1,m}(a_1,\ldots , a_{m})[b_2,\ldots , b_n,b_1' {\overline{\omega }}]\Vert _{L_2({\mathbb {S}})}\\&\quad \le C\Vert b_1\Vert _{H^{\tau }({\mathbb {S}})}\Vert {\overline{\omega }}\Vert _{H^1({\mathbb {S}})}\prod _{i=2}^n\Vert b_i\Vert _{H^r({\mathbb {S}})}. \end{aligned} \end{aligned}$$(A.2)In particular, \(C_{n,m}(a_{1}, \ldots , a_{m})\) has an extension in
$$\begin{aligned} {\mathcal {L}}_{n+1}(H^1({\mathbb {S}})\times (H^r({\mathbb {S}}))^{n-1}\times H^{\tau }({\mathbb {S}}), L_2({\mathbb {S}})). \end{aligned}$$ -
(ii2)
\(C_{n,m}\in \mathrm{C}^{1-}((H^r({\mathbb {S}}))^m,{\mathcal {L}}_{n+1}(H^1({\mathbb {S}})\times (H^{r}({\mathbb {S}}))^{n-1}\times H^{\tau }({\mathbb {S}}), L_2({\mathbb {S}}))).\)
-
(ii1)
-
(iii)
Let \(m,\, n\in {\mathbb {N}}\), \(r\in (3/2,2)\), and \(\tau \in (1/2,1).\) Then:
-
(iii1)
Given \(a_1,\ldots , a_m\in H^r({\mathbb {S}})\) and \( b_1,\ldots , b_n,\, {\overline{\omega }}\in \mathrm{C}^\infty ({\mathbb {S}}),\) there exists a constant C that depends only on n, m, r, \(\tau \), and \(\max _{i=1,\ldots , m}\Vert a_i\Vert _{H^r({\mathbb {S}})}\) such that
$$\begin{aligned}&\Vert C_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,{\overline{\omega }}]\Vert _{\infty }\le C\Vert {\overline{\omega }}\Vert _{H^\tau ({\mathbb {S}})} \prod _{i=1}^n\Vert b_i\Vert _{H^r({\mathbb {S}})}. \end{aligned}$$(A.3)In particular, \(C_{n,m}(a_{1}, \ldots , a_{m})\) has an extension in \( {\mathcal {L}}_{n+1}( (H^r({\mathbb {S}}))^{n}\times H^{\tau }({\mathbb {S}}), L_\infty ({\mathbb {S}})).\)
-
(iii2)
\(C_{n,m}\in \mathrm{C}^{1-}((H^r({\mathbb {S}}))^m,{\mathcal {L}}_{n+1}( (H^{r}({\mathbb {S}}))^{n}\times H^{\tau }({\mathbb {S}}), L_\infty ({\mathbb {S}}))).\)
-
(iii1)
Proof
We first address (i). To this end we fix \(\varphi \in \mathrm{C}^\infty _0({\mathbb {R}},[0,1])\) with \(\varphi =1\) for \(|x|\le 2\pi \) and \(\varphi =0\) for \(|x|\ge 4\pi \). Then, it is easy to see that
For \(|x|<\pi \) we have
and it follows from [48, Lemma 3.1] and (A.4) that
Moreover, it holds that
Herewith we established the estimate stated at (i). If \(a_1,\ldots , a_{m},\, b_1, \ldots , b_n \) are \( 2\pi \)-periodic, then so is also the function \(C_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n, {\overline{\omega }}]\), and the local Lipschitz continuity property of \(C_{n,m}\) follows directly from the estimate.
In order to prove (ii) we start by noticing that for \(h\in \mathrm{C}^\infty ({\mathbb {S}})\) it holds that
Using this relation we get
and the estimate established at (i) yields
We are left with the singular integral term
where
for \(x\in {\mathbb {R}}\) and \(s\ne 0\). The relation (A.6) is obtained by using integration by parts. We next estimate the terms on the right hand side of (A.6) separately. Firstly, it is easy to see that
Secondly, concerning the last two terms in (A.6), we may adapt the arguments from the nonperiodic case [48, Lemma 3.2], to arrive at
Indeed, since \(H^\tau ({\mathbb {S}})\hookrightarrow \mathrm{C}^{\tau -1/2}({\mathbb {S}})\), we obtain after appealing to Minkowski’s inequality thatFootnote 7
where, taking into account that \(|e^{ix}-1-ix|\le 2|x|^r\) for all \(x\in {\mathbb {R}},\) we have
Since \(r+\tau -7/2>-1\), the estimate (A.8)\(_1\) follows immediately (similarly for (A.8)\(_2.\))
Thirdly, for the remaining term
in (A.6) we obtain, in virtue of (i), that
and (A.2) follows from (A.7), (A.8), and (A.9).
In order to derive (A.1), we use the identity \(\partial (\delta _{[x,s]}{\overline{\omega }})/\partial s={\overline{\omega }}'(x-s)\) and integration by parts to recast T as
with
Concerning the integral terms in the last sum in (A.10), the embedding \(H^{1}({\mathbb {S}})\hookrightarrow \mathrm{C}^{r-3/2}({\mathbb {S}})\) together with Minkowski’s inequality yields
where we have used the relation \(|e^{ix}-1|\le C|x|^{\tau }\), \(x\in {\mathbb {R}},\) when deriving the fourth line.
Similarly, we find for \(2\le j\le n\) that
In the special case when \(j=1\), we use the procedure which led to (A.12) together with (i) to conclude that
The property (A.1) follows now from (A.5), (A.7), (A.8), and (A.11)–(A.14). The extension property left at (ii1) follows from (A.1). The claim (ii2) is a straight forward consequence of (A.1).
With respect to (iii) we decompose
with
Since \(\tau >1/2\) and \(H^{\tau }({\mathbb {S}})\hookrightarrow \mathrm{C}^{\tau -1/2}({\mathbb {S}})\), it holds
and we are left with the function A. Taking advantage of the embedding \(H^r({\mathbb {S}})\hookrightarrow \mathrm{C}^{r-1/2}({\mathbb {S}}),\) the arguments in the proof of [49, Lemma 3.1] show that indeed
The estimates (A.15), (A.16) lead us to the estimate (A.3). The last two claims follow directly from (A.3) and the proof is complete. \(\square \)
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Matioc, BV. Well-Posedness and Stability Results for Some Periodic Muskat Problems. J. Math. Fluid Mech. 22, 31 (2020). https://doi.org/10.1007/s00021-020-00494-7
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DOI: https://doi.org/10.1007/s00021-020-00494-7